From f3989de6132c5911989e9cbcd922f9541a1e94a2 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Wed, 16 Nov 2022 23:00:23 +0800 Subject: [PATCH] 20221116 night --- 工具/批量添加题库字段数据.ipynb | 450 ++++++----- 工具/文本文件/metadata.txt | 1170 ++++++++++++++++----------- 题库0.3/Problems.json | 1305 +++++++++++++++++++++++-------- 3 files changed, 1922 insertions(+), 1003 deletions(-) diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb index 0680eedd..4f64c059 100644 --- a/工具/批量添加题库字段数据.ipynb +++ b/工具/批量添加题库字段数据.ipynb @@ -2,241 +2,227 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.974\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t1.000\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.974\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.974\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.974\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.263\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.868\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.632\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.711\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.263\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.158\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.132\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.842\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.947\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.342\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.500\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.912\t0.671\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.917\t0.589\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三10班\t0.535\t0.247\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t1.000\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t1.000\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t1.000\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.952\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.952\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.143\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.762\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.714\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.952\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.191\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.191\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.048\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.857\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t1.000\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.286\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.238\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.809\t0.577\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.944\t0.566\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三11班\t0.436\t0.143\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.917\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.958\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.958\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t1.000\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.833\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.083\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.875\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.458\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.750\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.292\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.417\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.125\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.750\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t1.000\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.208\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.417\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.861\t0.776\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.917\t0.604\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三12班\t0.479\t0.292\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t1.000\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t1.000\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.969\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.969\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t1.000\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.938\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.969\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.969\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.969\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.844\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.938\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.812\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.938\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.938\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.875\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.875\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.964\t0.898\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.969\t0.945\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三01班\t0.911\t0.832\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.938\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.969\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.906\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.875\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.969\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.562\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.875\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.625\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.938\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.625\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.531\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.219\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t1.000\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.938\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.500\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.688\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.880\t0.871\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.880\t0.805\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三02班\t0.896\t0.609\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.935\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t1.000\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.903\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.968\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.903\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.516\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.806\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.742\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.871\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.516\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.613\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.258\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.903\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.935\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.645\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.613\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.806\t0.661\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.935\t0.726\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三03班\t0.892\t0.419\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t1.000\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.969\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.969\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.969\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.906\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.656\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.875\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.750\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.906\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.844\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.719\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.219\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.938\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t1.000\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.531\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.594\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.927\t0.844\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.958\t0.801\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三04班\t0.698\t0.469\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t1.000\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t1.000\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.974\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t1.000\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.974\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.421\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.895\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.737\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.947\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.421\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.553\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.368\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.895\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t1.000\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.474\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.658\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.961\t0.921\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.961\t0.796\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三05班\t0.864\t0.549\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.949\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.949\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.974\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.846\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t1.000\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.564\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.897\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.821\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.923\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.487\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.744\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.308\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.923\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.974\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.590\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.692\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.910\t0.779\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.944\t0.689\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三06班\t0.855\t0.657\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t1.000\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.971\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.912\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.912\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.882\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.176\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.882\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.735\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.735\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.324\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.235\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.088\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.853\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.971\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.324\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.382\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.971\t0.949\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.873\t0.551\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三07班\t0.475\t0.349\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.969\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t1.000\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.938\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.906\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.938\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.594\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.906\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.781\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.812\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.438\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.500\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.281\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.938\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.969\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.375\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.406\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.979\t0.961\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.938\t0.785\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三08班\t0.750\t0.379\n", - "题号: 005304 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.935\n", - "题号: 004468 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t1.000\n", - "题号: 004469 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.968\n", - "题号: 004470 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.871\n", - "题号: 004471 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.871\n", - "题号: 004472 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.419\n", - "题号: 004473 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.839\n", - "题号: 030480 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.677\n", - "题号: 004475 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.774\n", - "题号: 004476 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.258\n", - "题号: 004477 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.484\n", - "题号: 004478 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.194\n", - "题号: 004479 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.806\n", - "题号: 004480 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.968\n", - "题号: 004481 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.290\n", - "题号: 004482 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.452\n", - "题号: 004484 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.882\t0.677\n", - "题号: 004485 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.935\t0.661\n", - "题号: 004486 , 字段: usages 中已添加数据: 20221116\t2023届高三09班\t0.581\t0.258\n" + "题号: 000270 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 000271 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 000271 , 字段: objs 中已有该数据: K0714005X\n", + "题号: 000284 , 字段: objs 中已有该数据: K0715003X\n", + "题号: 000285 , 字段: objs 中已添加数据: K0714004X\n", + "题号: 000288 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 000379 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 000485 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 000485 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 000510 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 000560 , 字段: objs 中已添加数据: K0714003X\n", + "题号: 000625 , 字段: objs 中已添加数据: K0714004X\n", + "题号: 000625 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 000705 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 000723 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 000755 , 字段: objs 中已添加数据: K0714002X\n", + "题号: 000801 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 000813 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 000895 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 000908 , 字段: objs 中已添加数据: K0714003X\n", + "题号: 002306 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 002307 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002307 , 字段: objs 中已添加数据: K0714005X\n", + "题号: 002308 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002308 , 字段: objs 中已添加数据: K0714005X\n", + "题号: 002309 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002310 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002311 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 002313 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002314 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002315 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002316 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002317 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 002318 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 002319 , 字段: objs 中已添加数据: KNONE\n", + "题号: 002321 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002322 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002323 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002324 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002325 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 002326 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002327 , 字段: objs 中已添加数据: K0714003X\n", + "题号: 002327 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002328 , 字段: objs 中已添加数据: KNONE\n", + "题号: 002329 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 002330 , 字段: objs 中已添加数据: K0714003X\n", + "题号: 002330 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 002331 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002332 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002333 , 字段: objs 中已添加数据: K0714002X\n", + "题号: 002334 , 字段: objs 中已添加数据: K0713001X\n", + "题号: 002335 , 字段: objs 中已添加数据: K0714005X\n", + "题号: 002335 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 002337 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 002338 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002339 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 002342 , 字段: objs 中已添加数据: K0714005X\n", + "题号: 002342 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 002343 , 字段: objs 中已添加数据: K0714005X\n", + "题号: 002343 , 字段: objs 中已添加数据: K0714002X\n", + "题号: 002344 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 002345 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002346 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 002347 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 002356 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002363 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002367 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002380 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002389 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002428 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002428 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002443 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 002447 , 字段: objs 中已添加数据: KNONE\n", + "题号: 002469 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 002480 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 002481 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 002482 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 002683 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 002686 , 字段: objs 中已添加数据: KNONE\n", + "题号: 002688 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003399 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 003400 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 003401 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 003401 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 003402 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 003403 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003404 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003406 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 003407 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003408 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003409 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 003409 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003410 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 003411 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 003411 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 003413 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003414 , 字段: objs 中已添加数据: KNONE\n", + "题号: 003415 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 003416 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003428 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 003608 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003619 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003650 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003664 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 003704 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003733 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 003748 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 003826 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 003858 , 字段: objs 中已添加数据: K0714004X\n", + "题号: 003895 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 003909 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 003909 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 003918 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 004078 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 004099 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004120 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004129 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004152 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 004162 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 004162 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004182 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004204 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 004204 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004221 , 字段: objs 中已添加数据: K0713001X\n", + "题号: 004221 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 004242 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 004242 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004288 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 004288 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 004288 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004303 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 004309 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 004309 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004361 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004487 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 004524 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 004529 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 004529 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004561 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 004633 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 004701 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 004717 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004722 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 004722 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 004743 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 004743 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 008873 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 008873 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 008875 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008876 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008877 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008878 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008879 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 008879 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 008880 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 008881 , 字段: objs 中已添加数据: K0714003X\n", + "题号: 008882 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008883 , 字段: objs 中已添加数据: K0714003X\n", + "题号: 008884 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008884 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 008885 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 008886 , 字段: objs 中已添加数据: K0713001X\n", + "题号: 008887 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 008887 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 008889 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 008890 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 008890 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 008891 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 008892 , 字段: objs 中已添加数据: K0714003X\n", + "题号: 008892 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 008893 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 008894 , 字段: objs 中已添加数据: K0713001X\n", + "题号: 008895 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008897 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008897 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 008899 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 008916 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008916 , 字段: objs 中已添加数据: K0714003X\n", + "题号: 008945 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008946 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 008947 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 008948 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 008949 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 008959 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008962 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 008965 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 009080 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 009087 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 009089 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 009105 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 009109 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 009109 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 009819 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 009820 , 字段: objs 中已添加数据: K0714003X\n", + "题号: 009821 , 字段: objs 中已添加数据: K0714005X\n", + "题号: 009822 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 009823 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 009824 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 009825 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 010003 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 010003 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 010667 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 010668 , 字段: objs 中已添加数据: K0714003X\n", + "题号: 010668 , 字段: objs 中已添加数据: K0714005X\n", + "题号: 010669 , 字段: objs 中已添加数据: K0713004X\n", + "题号: 010670 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 010671 , 字段: objs 中已添加数据: K0713001X\n", + "题号: 010672 , 字段: objs 中已添加数据: K0715002X\n", + "题号: 010673 , 字段: objs 中已添加数据: K0713003X\n", + "题号: 010673 , 字段: objs 中已添加数据: K0715001X\n", + "题号: 010690 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 010702 , 字段: objs 中已添加数据: K0715003X\n", + "题号: 010703 , 字段: objs 中已添加数据: K0713002X\n", + "题号: 010704 , 字段: objs 中已添加数据: K0715003X\n" ] } ], @@ -334,7 +320,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -348,12 +334,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index 501a3e1d..d9fcc11d 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,686 +1,942 @@ -usages +objs +270 +K0713002X -005304 -20221116 2023届高三10班 0.974 -004468 -20221116 2023届高三10班 1.000 -004469 -20221116 2023届高三10班 0.974 +271 +K0713002X +K0714005X -004470 -20221116 2023届高三10班 0.974 -004471 -20221116 2023届高三10班 0.974 -004472 -20221116 2023届高三10班 0.263 +284 +K0715003X -004473 -20221116 2023届高三10班 0.868 -030480 -20221116 2023届高三10班 0.632 -004475 -20221116 2023届高三10班 0.711 +285 +K0714004X -004476 -20221116 2023届高三10班 0.263 -004477 -20221116 2023届高三10班 0.158 -004478 -20221116 2023届高三10班 0.132 +288 +K0713002X -004479 -20221116 2023届高三10班 0.842 -004480 -20221116 2023届高三10班 0.947 -004481 -20221116 2023届高三10班 0.342 +379 +K0713002X -004482 -20221116 2023届高三10班 0.500 -004484 -20221116 2023届高三10班 0.912 0.671 -004485 -20221116 2023届高三10班 0.917 0.589 +485 +K0713002X +K0713004X -004486 -20221116 2023届高三10班 0.535 0.247 -005304 -20221116 2023届高三11班 1.000 -004468 -20221116 2023届高三11班 1.000 +510 +K0715002X -004469 -20221116 2023届高三11班 1.000 -004470 -20221116 2023届高三11班 0.952 -004471 -20221116 2023届高三11班 0.952 +560 +K0714003X -004472 -20221116 2023届高三11班 0.143 -004473 -20221116 2023届高三11班 0.762 -030480 -20221116 2023届高三11班 0.714 +625 +K0714004X +K0713002X -004475 -20221116 2023届高三11班 0.952 -004476 -20221116 2023届高三11班 0.191 -004477 -20221116 2023届高三11班 0.191 +705 +K0715002X -004478 -20221116 2023届高三11班 0.048 -004479 -20221116 2023届高三11班 0.857 -004480 -20221116 2023届高三11班 1.000 +723 +K0713003X -004481 -20221116 2023届高三11班 0.286 -004482 -20221116 2023届高三11班 0.238 -004484 -20221116 2023届高三11班 0.809 0.577 +755 +K0714002X -004485 -20221116 2023届高三11班 0.944 0.566 -004486 -20221116 2023届高三11班 0.436 0.143 -005304 -20221116 2023届高三12班 0.917 +801 +K0713002X -004468 -20221116 2023届高三12班 0.958 -004469 -20221116 2023届高三12班 0.958 -004470 -20221116 2023届高三12班 1.000 +813 +K0715003X -004471 -20221116 2023届高三12班 0.833 -004472 -20221116 2023届高三12班 0.083 -004473 -20221116 2023届高三12班 0.875 +895 +K0715002X -030480 -20221116 2023届高三12班 0.458 -004475 -20221116 2023届高三12班 0.750 -004476 -20221116 2023届高三12班 0.292 +908 +K0714003X -004477 -20221116 2023届高三12班 0.417 -004478 -20221116 2023届高三12班 0.125 -004479 -20221116 2023届高三12班 0.750 +2306 +K0713004X -004480 -20221116 2023届高三12班 1.000 -004481 -20221116 2023届高三12班 0.208 -004482 -20221116 2023届高三12班 0.417 +2307 +K0713002X +K0714005X -004484 -20221116 2023届高三12班 0.861 0.776 -004485 -20221116 2023届高三12班 0.917 0.604 -004486 -20221116 2023届高三12班 0.479 0.292 +2308 +K0713002X +K0714005X -005304 -20221116 2023届高三01班 1.000 -004468 -20221116 2023届高三01班 1.000 -004469 -20221116 2023届高三01班 0.969 +2309 +K0713002X -004470 -20221116 2023届高三01班 0.969 -004471 -20221116 2023届高三01班 1.000 -004472 -20221116 2023届高三01班 0.938 +2310 +K0713002X -004473 -20221116 2023届高三01班 0.969 -030480 -20221116 2023届高三01班 0.969 -004475 -20221116 2023届高三01班 0.969 +2311 +K0713004X -004476 -20221116 2023届高三01班 0.844 -004477 -20221116 2023届高三01班 0.938 -004478 -20221116 2023届高三01班 0.812 +2313 +K0713002X -004479 -20221116 2023届高三01班 0.938 -004480 -20221116 2023届高三01班 0.938 -004481 -20221116 2023届高三01班 0.875 +2314 +K0713002X -004482 -20221116 2023届高三01班 0.875 -004484 -20221116 2023届高三01班 0.964 0.898 -004485 -20221116 2023届高三01班 0.969 0.945 +2315 +K0713002X -004486 -20221116 2023届高三01班 0.911 0.832 -005304 -20221116 2023届高三02班 0.938 -004468 -20221116 2023届高三02班 0.969 +2316 +K0713002X -004469 -20221116 2023届高三02班 0.906 -004470 -20221116 2023届高三02班 0.875 -004471 -20221116 2023届高三02班 0.969 +2317 +K0713004X -004472 -20221116 2023届高三02班 0.562 -004473 -20221116 2023届高三02班 0.875 -030480 -20221116 2023届高三02班 0.625 +2318 +K0713004X -004475 -20221116 2023届高三02班 0.938 -004476 -20221116 2023届高三02班 0.625 -004477 -20221116 2023届高三02班 0.531 +2319 +KNONE -004478 -20221116 2023届高三02班 0.219 -004479 -20221116 2023届高三02班 1.000 -004480 -20221116 2023届高三02班 0.938 +2321 +K0715002X -004481 -20221116 2023届高三02班 0.500 -004482 -20221116 2023届高三02班 0.688 -004484 -20221116 2023届高三02班 0.880 0.871 +2322 +K0715002X -004485 -20221116 2023届高三02班 0.880 0.805 -004486 -20221116 2023届高三02班 0.896 0.609 -005304 -20221116 2023届高三03班 0.935 +2323 +K0713002X -004468 -20221116 2023届高三03班 1.000 -004469 -20221116 2023届高三03班 0.903 -004470 -20221116 2023届高三03班 0.968 +2324 +K0715002X -004471 -20221116 2023届高三03班 0.903 -004472 -20221116 2023届高三03班 0.516 -004473 -20221116 2023届高三03班 0.806 +2325 +K0715003X -030480 -20221116 2023届高三03班 0.742 -004475 -20221116 2023届高三03班 0.871 -004476 -20221116 2023届高三03班 0.516 +2326 +K0715002X -004477 -20221116 2023届高三03班 0.613 -004478 -20221116 2023届高三03班 0.258 -004479 -20221116 2023届高三03班 0.903 +2327 +K0714003X +K0713002X -004480 -20221116 2023届高三03班 0.935 -004481 -20221116 2023届高三03班 0.645 -004482 -20221116 2023届高三03班 0.613 +2328 +KNONE -004484 -20221116 2023届高三03班 0.806 0.661 -004485 -20221116 2023届高三03班 0.935 0.726 -004486 -20221116 2023届高三03班 0.892 0.419 +2329 +K0713003X -005304 -20221116 2023届高三04班 1.000 -004468 -20221116 2023届高三04班 0.969 -004469 -20221116 2023届高三04班 0.969 +2330 +K0714003X +K0713004X -004470 -20221116 2023届高三04班 0.969 -004471 -20221116 2023届高三04班 0.906 -004472 -20221116 2023届高三04班 0.656 +2331 +K0713002X -004473 -20221116 2023届高三04班 0.875 -030480 -20221116 2023届高三04班 0.750 -004475 -20221116 2023届高三04班 0.906 +2332 +K0715002X -004476 -20221116 2023届高三04班 0.844 -004477 -20221116 2023届高三04班 0.719 -004478 -20221116 2023届高三04班 0.219 +2333 +K0714002X -004479 -20221116 2023届高三04班 0.938 -004480 -20221116 2023届高三04班 1.000 -004481 -20221116 2023届高三04班 0.531 +2334 +K0713001X -004482 -20221116 2023届高三04班 0.594 -004484 -20221116 2023届高三04班 0.927 0.844 -004485 -20221116 2023届高三04班 0.958 0.801 +2335 +K0714005X +K0713004X -004486 -20221116 2023届高三04班 0.698 0.469 -005304 -20221116 2023届高三05班 1.000 -004468 -20221116 2023届高三05班 1.000 +2337 +K0715003X -004469 -20221116 2023届高三05班 0.974 -004470 -20221116 2023届高三05班 1.000 -004471 -20221116 2023届高三05班 0.974 +2338 +K0715002X -004472 -20221116 2023届高三05班 0.421 -004473 -20221116 2023届高三05班 0.895 -030480 -20221116 2023届高三05班 0.737 +2339 +K0715003X -004475 -20221116 2023届高三05班 0.947 -004476 -20221116 2023届高三05班 0.421 -004477 -20221116 2023届高三05班 0.553 +2342 +K0714005X +K0713004X -004478 -20221116 2023届高三05班 0.368 -004479 -20221116 2023届高三05班 0.895 -004480 -20221116 2023届高三05班 1.000 +2343 +K0714005X +K0714002X -004481 -20221116 2023届高三05班 0.474 -004482 -20221116 2023届高三05班 0.658 -004484 -20221116 2023届高三05班 0.961 0.921 +2344 +K0715003X -004485 -20221116 2023届高三05班 0.961 0.796 -004486 -20221116 2023届高三05班 0.864 0.549 -005304 -20221116 2023届高三06班 0.949 +2345 +K0715002X -004468 -20221116 2023届高三06班 0.949 -004469 -20221116 2023届高三06班 0.974 -004470 -20221116 2023届高三06班 0.846 +2346 +K0715003X -004471 -20221116 2023届高三06班 1.000 -004472 -20221116 2023届高三06班 0.564 -004473 -20221116 2023届高三06班 0.897 +2347 +K0715003X -030480 -20221116 2023届高三06班 0.821 -004475 -20221116 2023届高三06班 0.923 -004476 -20221116 2023届高三06班 0.487 +2356 +K0713002X -004477 -20221116 2023届高三06班 0.744 -004478 -20221116 2023届高三06班 0.308 -004479 -20221116 2023届高三06班 0.923 +2363 +K0713002X -004480 -20221116 2023届高三06班 0.974 -004481 -20221116 2023届高三06班 0.590 -004482 -20221116 2023届高三06班 0.692 +2367 +K0715002X -004484 -20221116 2023届高三06班 0.910 0.779 -004485 -20221116 2023届高三06班 0.944 0.689 -004486 -20221116 2023届高三06班 0.855 0.657 +2380 +K0713002X -005304 -20221116 2023届高三07班 1.000 -004468 -20221116 2023届高三07班 0.971 -004469 -20221116 2023届高三07班 0.912 +2389 +K0715002X -004470 -20221116 2023届高三07班 0.912 -004471 -20221116 2023届高三07班 0.882 -004472 -20221116 2023届高三07班 0.176 +2428 +K0713002X +K0715002X -004473 -20221116 2023届高三07班 0.882 -030480 -20221116 2023届高三07班 0.735 -004475 -20221116 2023届高三07班 0.735 +2443 +K0713004X -004476 -20221116 2023届高三07班 0.324 -004477 -20221116 2023届高三07班 0.235 -004478 -20221116 2023届高三07班 0.088 +2447 +KNONE -004479 -20221116 2023届高三07班 0.853 -004480 -20221116 2023届高三07班 0.971 -004481 -20221116 2023届高三07班 0.324 +2469 +K0713002X -004482 -20221116 2023届高三07班 0.382 -004484 -20221116 2023届高三07班 0.971 0.949 -004485 -20221116 2023届高三07班 0.873 0.551 -004486 -20221116 2023届高三07班 0.475 0.349 -005304 -20221116 2023届高三08班 0.969 -004468 -20221116 2023届高三08班 1.000 -004469 -20221116 2023届高三08班 0.938 -004470 -20221116 2023届高三08班 0.906 -004471 -20221116 2023届高三08班 0.938 +2480 +K0715003X -004472 -20221116 2023届高三08班 0.594 -004473 -20221116 2023届高三08班 0.906 -030480 -20221116 2023届高三08班 0.781 +2481 +K0715003X -004475 -20221116 2023届高三08班 0.812 -004476 -20221116 2023届高三08班 0.438 -004477 -20221116 2023届高三08班 0.500 +2482 +K0715003X -004478 -20221116 2023届高三08班 0.281 -004479 -20221116 2023届高三08班 0.938 -004480 -20221116 2023届高三08班 0.969 +2683 +K0715002X -004481 -20221116 2023届高三08班 0.375 -004482 -20221116 2023届高三08班 0.406 -004484 -20221116 2023届高三08班 0.979 0.961 +2686 +KNONE -004485 -20221116 2023届高三08班 0.938 0.785 -004486 -20221116 2023届高三08班 0.750 0.379 -005304 -20221116 2023届高三09班 0.935 +2688 +K0715003X -004468 -20221116 2023届高三09班 1.000 -004469 -20221116 2023届高三09班 0.968 -004470 -20221116 2023届高三09班 0.871 +3399 +K0713002X -004471 -20221116 2023届高三09班 0.871 -004472 -20221116 2023届高三09班 0.419 -004473 -20221116 2023届高三09班 0.839 +3400 +K0713002X -030480 -20221116 2023届高三09班 0.677 -004475 -20221116 2023届高三09班 0.774 -004476 -20221116 2023届高三09班 0.258 +3401 +K0713003X +K0713004X -004477 -20221116 2023届高三09班 0.484 -004478 -20221116 2023届高三09班 0.194 -004479 -20221116 2023届高三09班 0.806 +3402 +K0715002X -004480 -20221116 2023届高三09班 0.968 -004481 -20221116 2023届高三09班 0.290 -004482 -20221116 2023届高三09班 0.452 +3403 +K0715003X + + + +3404 +K0715003X + + + +3406 +K0715002X + + + +3407 +K0715003X + + + +3408 +K0715003X + + + +3409 +K0713002X +K0715003X + + + +3410 +K0713003X + + + +3411 +K0713002X +K0713004X + + + +3413 +K0715003X + + + +3414 +KNONE + + + +3415 +K0715002X + + + +3416 +K0715003X + + + + + + +3428 +K0715002X + + + +3608 +K0715003X + + + +3619 +K0715003X + + + +3650 +K0715003X + + + +3664 +K0713002X + + + + + + + + + +3704 +K0715003X + + + +3733 +K0713003X + + + +3748 +K0713002X + + + +3826 +K0713002X + + + +3858 +K0714004X + + + +3895 +K0713002X + + + +3909 +K0713003X +K0715003X + + + +3918 +K0713003X + + + +4078 +K0713003X + + + +4099 +K0715003X + + + +4120 +K0715003X + + + +4129 +K0715003X + + + +4152 +K0713002X + + + +4162 +K0713004X +K0715003X + + + +4182 +K0715003X + + + +4204 +K0713003X +K0715003X + + + +4221 +K0713001X +K0713002X + + + +4242 +K0713002X +K0715003X + + + +4288 +K0713003X +K0715002X +K0715003X + + + +4303 +K0713002X + + + +4309 +K0713002X +K0715003X + + + +4361 +K0715003X + + + +4487 +K0713002X + + + +4524 +K0713003X + + + +4529 +K0713002X +K0715003X + + + +4561 +K0713002X + + + +4633 +K0713002X + + + +4701 +K0713003X + + + +4717 +K0715003X + + + +4722 +K0713004X +K0715003X + + + +4743 +K0713003X +K0715003X + + + +8873 +K0713003X +K0713004X + + + +8875 +K0713002X + + + +8876 +K0713002X + + + +8877 +K0713002X + + + +8878 +K0713002X + + + +8879 +K0713003X +K0713004X + + + +8880 +K0713004X + + + +8881 +K0714003X + + + +8882 +K0713002X + + + +8883 +K0714003X + + + +8884 +K0713002X +K0713004X + + + +8885 +K0715003X + + + +8886 +K0713001X + + + +8887 +K0713003X +K0713004X + + + +8889 +K0713003X + + + +8890 +K0713003X +K0715003X + + + +8891 +K0715002X + + + +8892 +K0714003X +K0713003X + + + +8893 +K0715002X + + + +8894 +K0713001X + + + +8895 +K0713002X + + + +8897 +K0713002X +K0713003X + + + +8899 +K0715003X + + + +8916 +K0713002X +K0714003X + + + +8945 +K0713002X + + + +8946 +K0715003X + + + +8947 +K0715003X + + + +8948 +K0715003X + + + +8949 +K0715003X + + + +8959 +K0713002X + + + +8962 +K0713002X + + + +8965 +K0715002X + + + +9080 +K0713002X + + + +9087 +K0715003X + + + +9089 +K0715003X + + + +9105 +K0713002X + + + +9109 +K0713004X +K0715003X + + + +9819 +K0713004X + + + +9820 +K0714003X + + + +9821 +K0714005X + + + +9822 +K0713003X + + + +9823 +K0715002X + + + +9824 +K0715002X + + + +9825 +K0715003X + + + +10003 +K0713003X +K0715003X + + + +10667 +K0713002X + + + +10668 +K0714003X +K0714005X + + + +10669 +K0713004X + + + +10670 +K0715003X + + + +10671 +K0713001X + + + +10672 +K0715002X + + + +10673 +K0713003X +K0715001X + + + +10690 +K0713002X + + + + + + +10702 +K0715003X + + + +10703 +K0713002X + + + +10704 +K0715003X -004484 -20221116 2023届高三09班 0.882 0.677 -004485 -20221116 2023届高三09班 0.935 0.661 -004486 -20221116 2023届高三09班 0.581 0.258 diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 9bc2e9c5..56647d90 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -6975,7 +6975,8 @@ "content": "对圆$(x-a)^2+(y+b)^2=a^2+b^2\\ (a>0, \\ b>0)$, 下列说法是否正确, 请说明理由:\\\\\n(1) 该圆的圆心为$(a, b)$;\\\\\n(2) 该圆过原点;\\\\\n(3) 该圆与$x$轴相交于两个不同点.", "objs": [ "K0709003X", - "K0711002X" + "K0711002X", + "K0709001X" ], "tags": [ "第七单元", @@ -6999,7 +7000,8 @@ "id": "000270", "content": "若椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{a^2}=1$与双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}2=1$有相同的焦点, 求实数$a$的值.", "objs": [ - "K0713004X" + "K0713004X", + "K0713002X" ], "tags": [ "第七单元", @@ -7026,7 +7028,8 @@ "id": "000271", "content": "设椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\ (a>b>0)$的焦距为$2c$. 若$b^2=ac$, 求该椭圆的离心率.", "objs": [ - "K0714005X" + "K0714005X", + "K0713002X" ], "tags": [ "第七单元", @@ -7380,7 +7383,8 @@ "id": "000285", "content": "已知定点$A(a, 0) \\ (00)$上的一点$P$也在抛物线$y^2=\\dfrac94x$上, 抛物线焦点为$F_3$, 若$|PF_3|=\\dfrac{25}{16}$, 则$\\triangle PF_1F_2$的面积为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X", + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -13559,7 +13573,9 @@ "000501": { "id": "000501", "content": "过点$P(-2,1)$作圆$x^2+y^2=5$的切线, 则该切线的点法向式方程是\\blank{50}.", - "objs": [], + "objs": [ + "K0711003X" + ], "tags": [ "第七单元", "圆" @@ -13800,7 +13816,9 @@ "000510": { "id": "000510", "content": "已知$F_1$、$F_2$是椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$的两个焦点, $P$是椭圆上的一个动点, 则$|PF_1|\\times |PF_2|$的最大值是\\blank{50}.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -14940,7 +14958,9 @@ "000553": { "id": "000553", "content": "若直线$ax-y+3=0$与圆$(x-1)^2+(y-2)^2=4$相交于$A$、$B$两点, 且$|AB|=2 \\sqrt3$, 则$a$=\\blank{50}.", - "objs": [], + "objs": [ + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -15131,7 +15151,9 @@ "000560": { "id": "000560", "content": "在平面直角坐标系$xOy$中, 以直线$y=\\pm 2x$为渐近线, 且经过椭圆$x^2+\\dfrac{y^2}4=1$右顶点的双曲线的方程是\\blank{50}.", - "objs": [], + "objs": [ + "K0714003X" + ], "tags": [ "第七单元", "椭圆", @@ -15391,7 +15413,9 @@ "000570": { "id": "000570", "content": "已知圆$O:x^2+y^2=1$与圆$O'$关于直线$x+y=5$对称, 则圆$O'$的方程是\\blank{50}.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -16834,7 +16858,10 @@ "000625": { "id": "000625", "content": "已知椭圆$x^2+\\dfrac{y^2}{b^2}=1\\ (00$), 若圆$C$上至少存在一点$P$, 使得$\\angle APB=90^\\circ $, 则$m$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0712002X" + ], "tags": [ "第七单元", "圆" @@ -18980,7 +19009,9 @@ "000705": { "id": "000705", "content": "设$A$是椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{a^2-4}=1 \\ (a>0)$上的动点, 点$F$的坐标为$(-2,0)$, 若满足$|AF|=10$的点$A$有且仅有两个, 则实数$a$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -19460,7 +19491,9 @@ "000723": { "id": "000723", "content": "已知椭圆$\\dfrac{x^2}{a^2}+y^2=1 \\ (a>0)$的焦点$F_1$、$F_2$, 抛物线${y^2}=2x$的焦点为$F$, 若$\\overrightarrow{F_1F}=3 \\overrightarrow{FF_2}$, 则$a=$\\blank{50}.", - "objs": [], + "objs": [ + "K0713003X" + ], "tags": [ "第七单元", "椭圆" @@ -20036,7 +20069,9 @@ "000745": { "id": "000745", "content": "已知直线$l_1:mx-y=0$, $l_2:x+my-m-2=0$. 当$m$在实数范围内变化时, $l_1$与$l_2$的交点$P$恒在一个定圆上, 则定圆方程是\\blank{50}.", - "objs": [], + "objs": [ + "K0709007X" + ], "tags": [ "第七单元", "直线", @@ -20293,7 +20328,9 @@ "000755": { "id": "000755", "content": "椭圆的长轴长等于$m$, 短轴长等于$n$, 则此椭圆的内接矩形的面积的最大值为\\blank{50}.", - "objs": [], + "objs": [ + "K0714002X" + ], "tags": [ "第七单元", "椭圆" @@ -21565,7 +21602,9 @@ "000801": { "id": "000801", "content": "椭圆$\\begin{cases} x=2 \\cos\\theta, \\\\ y=\\sqrt3\\sin\\theta \\end{cases}$($\\theta$为参数)的右焦点为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆", @@ -21866,7 +21905,9 @@ "000813": { "id": "000813", "content": "在平面直角坐标系$xOy$中, 直线$l$的参数方程为$\\begin{cases} x=\\dfrac{\\sqrt2}2t-\\sqrt2, \\\\ y=\\dfrac{\\sqrt2}4t, \\end{cases}$($t$为参数), 椭圆$C$的参数方程为$\\begin{cases} x=\\cos \\theta, \\\\ y=\\dfrac12\\sin \\theta, \\end{cases}$($\\theta$为参数), 则直线$l$与椭圆$C$的公共点坐标为\\blank{50}.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆", @@ -24054,7 +24095,9 @@ "000895": { "id": "000895", "content": "已知$F_1,F_2$是椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1\\ (a>b>0)$的两个焦点, $P$为椭圆上一点, 且$\\overrightarrow{PF_1}\\perp \\overrightarrow{PF_2}$, 若$\\triangle PF_1F_2$的面积为$9$, 则$b=$\\blank{50}.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -24388,7 +24431,9 @@ "000908": { "id": "000908", "content": "如图, $A$、$B$为椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\ (a>b>0)$的两个顶点, 过椭圆的右焦点$F$作$x$轴的垂线, 与其交于点$C$. 若$AB\\parallel OC$($O$为坐标原点), 则直线$AB$的斜率为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[scale = 1.4]\n \\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n \\draw [->] (0,-1.4) -- (0,1.4) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,0) ellipse ({sqrt(2)} and 1);\n \\draw (1,0) node [below] {$F$} -- (1,{sqrt(2)/2}) node [above] {$C$} -- (0,0);\n \\draw ({-sqrt(2)},0) node [below left] {$A$} -- (0,1) node [above right] {$B$};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0714003X" + ], "tags": [ "第七单元", "椭圆" @@ -24736,7 +24781,9 @@ "000922": { "id": "000922", "content": "圆$C:x^2+y^2-2x-4y+4=0$的圆心到直线$3x+4y+4=0$的距离$d=$\\blank{50}.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -25227,7 +25274,9 @@ "000941": { "id": "000941", "content": "若$AB$是圆$x^2+(y-3)^2=1$的任意一条直径, $O$为坐标原点, 则$\\overrightarrow{OA}\\cdot \\overrightarrow{OB}$的值为\\blank{50}.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -57015,7 +57064,10 @@ "002118": { "id": "002118", "content": "已知圆$O$以原点为圆心, $5$为半径, 设$A(3,4),B(0,-5)$, 则\\\\ \n(1) 劣弧$\\overset\\frown{AB}$的方程为\\blank{150};\\\\ \n(2) 优弧$\\overset\\frown{AB}$的方程为\\blank{150}.", - "objs": [], + "objs": [ + "K0709003X", + "K0709005X" + ], "tags": [ "第七单元", "圆" @@ -60349,7 +60401,9 @@ "002239": { "id": "002239", "content": "已知圆$(x-a)^2+(y-b)^2=r^2(r>0)$, 写出在下列情况下, $a,b,r$分别应满足的条件.\\\\ \n(1) 圆过原点; \\blank{100};\\\\ \n(2) 圆心在$x$轴上; \\blank{100};\\\\ \n(3) 圆与$x$轴相切; \\blank{100};\\\\ \n(4) 圆与两坐标轴都相切; \\blank{100}.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -60374,7 +60428,9 @@ "002240": { "id": "002240", "content": "以$A(-1,2)$, $B(5,-6)$为直径两端点的圆的一般方程为\\blank{100}.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -60399,7 +60455,9 @@ "002241": { "id": "002241", "content": "过点$M(5,2)$, $N(3,2)$, 且圆心在直线$y=2x-3$上的圆的标准方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -60426,7 +60484,10 @@ "002242": { "id": "002242", "content": "过点$P(-8,-1)$, $Q(5,12)$, $R(17,4)$的圆的圆心坐标为\\blank{50}.", - "objs": [], + "objs": [ + "K0710004X", + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -60451,7 +60512,9 @@ "002243": { "id": "002243", "content": "已知$D^2+E^2>4F$, 圆$x^2+y^2+Dx+Ey+F=0$关于直线$y=x$对称的充分必要条件是\\blank{50}.", - "objs": [], + "objs": [ + "K0710001X" + ], "tags": [ "第七单元", "圆" @@ -60476,7 +60539,9 @@ "002244": { "id": "002244", "content": "圆$x^2+y^2-4y=0$关于直线$x-y+1=0$对称所得的圆的一般方程是\\blank{50}.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -60501,7 +60566,9 @@ "002245": { "id": "002245", "content": "已知$P(3,0)$是圆$x^2+y^2-8x-2y+12=0$内一点, 在过点$P$的弦中, 最长的弦所在的直线方程是\\blank{80}, 最短的弦所在的直线方程是\\blank{80}.", - "objs": [], + "objs": [ + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -60550,7 +60617,9 @@ "002247": { "id": "002247", "content": "``$A=C\\neq0, B=0$''是``$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$表示圆的方程''的\\bracket{20}.\n\\twoch{充要条件}{充分非必要条件}{必要非充分条件}{既非充分又非必要条件}", - "objs": [], + "objs": [ + "K0710002X" + ], "tags": [ "第七单元", "圆" @@ -60625,7 +60694,9 @@ "002250": { "id": "002250", "content": "若方程$a^2x^2+(2a+3)y^2+2ax+a+1=0$表示圆, 则实数$a$的值为\\blank{50}.", - "objs": [], + "objs": [ + "K0710002X" + ], "tags": [ "第七单元", "圆" @@ -60652,7 +60723,9 @@ "002251": { "id": "002251", "content": "已知方程$x^2+y^2+2kx+4y+3k+8=0$表示一个圆, 则实数$k$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0710002X" + ], "tags": [ "第七单元", "圆" @@ -60677,7 +60750,10 @@ "002252": { "id": "002252", "content": "设平面上有两定点$A,B$, 动点$P$满足$\\dfrac{|PA|}{|PB|}=k$, 其中$k$为不等于$1$的正常数. 求证: $P$的轨迹是圆.", - "objs": [], + "objs": [ + "K0721002X", + "K0721004X" + ], "tags": [ "第七单元", "圆" @@ -60726,7 +60802,9 @@ "002254": { "id": "002254", "content": "已知圆$x^2+y^2+mx+ny+p=0$与$x$轴相切于原点, 则$m,n,p$应满足\\bracket{20}.\n\\twoch{$mn\\ne 0$且$p=0$}{$m\\ne 0$且$n^2+p^2=0$}{$n\\ne 0$且$m^2+p^2=0$}{$p\\ne 0$且$m^2+n^2=0$}", - "objs": [], + "objs": [ + "K0710001X" + ], "tags": [ "第七单元", "圆" @@ -60751,7 +60829,9 @@ "002255": { "id": "002255", "content": "若圆$x^2+y^2+4x+2by+b^2=0$与两坐标轴都相切, 那么$b$的值所组成的集合是\\blank{50}.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -60776,7 +60856,9 @@ "002256": { "id": "002256", "content": "圆心在直线$2x-y=3$上, 且与两坐标轴都相切的圆的一般方程为\\blank{100}.", - "objs": [], + "objs": [ + "K0710001X" + ], "tags": [ "第七单元", "圆" @@ -60801,7 +60883,9 @@ "002257": { "id": "002257", "content": "过点$P(3,4)$, 且与圆$x^2+y^2=25$相切的切线方程为\\blank{100}.", - "objs": [], + "objs": [ + "K0711003X" + ], "tags": [ "第七单元", "圆" @@ -60826,7 +60910,9 @@ "002258": { "id": "002258", "content": "和直线$3x-2y+4=0$垂直, 且和圆$x^2-2x+y^2-3=0$相切的直线方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0711004X" + ], "tags": [ "第七单元", "圆" @@ -60851,7 +60937,9 @@ "002259": { "id": "002259", "content": "直线$2x-y-1=0$被圆$x^2+y^2-2y-1=0$所截得的弦的长度为\\blank{50}.", - "objs": [], + "objs": [ + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -60876,7 +60964,9 @@ "002260": { "id": "002260", "content": "直线$l$过点$P(0,2)$, 且被圆$x^2+y^2=4$截得的弦长为$2$, 则$l$的方程为\\blank{100}.", - "objs": [], + "objs": [ + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -60901,7 +60991,9 @@ "002261": { "id": "002261", "content": "圆$x^2+y^2-4x-5=0$的弦$AB$以点$P(3,1)$为中点, 则直线$AB$的方程是\\blank{50}.", - "objs": [], + "objs": [ + "K0710006X" + ], "tags": [ "第七单元", "圆" @@ -60926,7 +61018,9 @@ "002262": { "id": "002262", "content": "圆$(x-3)^2+(y-3)^2=9$上到直线$3x+4y-11=0$的距离等于$1$的点有\\blank{40}个.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -60951,7 +61045,9 @@ "002263": { "id": "002263", "content": "使圆$(x-2)^2+(y+3)^2=2$上点与点$(0,-5)$距离最大的点的坐标是\\blank{50}.", - "objs": [], + "objs": [ + "K0712002X" + ], "tags": [ "第七单元", "圆" @@ -60976,7 +61072,9 @@ "002264": { "id": "002264", "content": "圆心在直线$4x+y=0$上, 且与直线$x+y-1=0$切于点$P(3,-2)$的圆的一般方程为\\blank{100}.", - "objs": [], + "objs": [ + "K0710004X" + ], "tags": [ "第七单元", "圆" @@ -61001,7 +61099,9 @@ "002265": { "id": "002265", "content": "和直线$x-6y-10=0$相切于点$Q(4,-1)$, 且经过点$M(9,6)$的圆的一般方程为\\blank{100}.", - "objs": [], + "objs": [ + "K0710004X" + ], "tags": [ "第七单元", "圆" @@ -61026,7 +61126,9 @@ "002266": { "id": "002266", "content": "过点$(2,-1)$, 圆心在直线$2x+y=0$上, 且与直线$x-y-1=0$相切的圆的方程为\\blank{100}.", - "objs": [], + "objs": [ + "K0710004X" + ], "tags": [ "第七单元", "圆" @@ -61053,7 +61155,9 @@ "002267": { "id": "002267", "content": "过点$M(3,0)$作直线$l$与圆$x^2+y^2=16$相交于$A,B$两点, 求$l$的方程, 使得$\\triangle AOB$的面积最大, 并求此最大值($O$为坐标原点).", - "objs": [], + "objs": [ + "K0710006X" + ], "tags": [ "第七单元", "圆" @@ -61080,7 +61184,9 @@ "002268": { "id": "002268", "content": "过点$M(3,0)$作直线$l$与圆$x^2+y^2=16$相交于$A,B$两点, 求$l$的方程, 使得$\\triangle AOB$的面积最大, 并求此最大值($O$为坐标原点).", - "objs": [], + "objs": [ + "K0710006X" + ], "tags": [ "第七单元", "圆" @@ -61107,7 +61213,9 @@ "002269": { "id": "002269", "content": "已知点$P(x_0,y_0)$在圆$x^2+y^2=r^2$外, 则直线$x_0x+y_0y=r^2$与该圆的位置关系为\\bracket{20}.\n\\fourch{相切}{相离}{相交}{不确定}", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -61132,7 +61240,9 @@ "002270": { "id": "002270", "content": "直线$ax=by$与圆$x^2+y^2-ax+by=0$的位置关系为\\bracket{20}.\n\\fourch{相切}{相离}{相交}{不确定}", - "objs": [], + "objs": [ + "K0711001X" + ], "tags": [ "第七单元", "圆" @@ -61185,7 +61295,9 @@ "002272": { "id": "002272", "content": "与圆$x^2+y^2=25$外切于点$P(4,-3)$, 且半径为$1$的圆的方程为\\blank{100}.", - "objs": [], + "objs": [ + "K0712002X" + ], "tags": [ "第七单元", "圆" @@ -61215,7 +61327,9 @@ "002273": { "id": "002273", "content": "自点$M(2,3)$向圆$x^2+y^2=1$引切线, 则切线长等于\\blank{50}.", - "objs": [], + "objs": [ + "K0712005X" + ], "tags": [ "第七单元", "圆" @@ -61240,7 +61354,10 @@ "002274": { "id": "002274", "content": "已知圆心在直线$x-3y=0$上的圆$C$与$y$轴相切, 且在直线$y=x$上截得的弦长为$2\\sqrt{7}$, 则该圆的一般方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0709006X", + "K0710004X" + ], "tags": [ "第七单元", "圆" @@ -61265,7 +61382,9 @@ "002275": { "id": "002275", "content": "自点$P(-3,3)$发出的光线$l$经$x$轴反射, 其反射线所在直线恰好与圆$x^2+y^2-4x-4y+7=0$相切, 则入射光线$l$所在直线的方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -61290,7 +61409,9 @@ "002276": { "id": "002276", "content": "已知直线$l:y=k(x+1)+n$, 若不论$k$取何值, $l$总与圆${{(x+3)}^{2}}+{{(y-3)}^{2}}=6$有公共点,\n则常数$n$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -61315,7 +61436,9 @@ "002277": { "id": "002277", "content": "求过$M(3,4)$且与圆$x^2+y^2=9$相切的直线的方程.", - "objs": [], + "objs": [ + "K0711004X" + ], "tags": [ "第七单元", "圆" @@ -61343,7 +61466,9 @@ "002278": { "id": "002278", "content": "设$m$是常数, 讨论圆$(x-m)^2+y^2=1$与圆$x^2+(y-m)^2=4$的位置关系.", - "objs": [], + "objs": [ + "K0712002X" + ], "tags": [ "第七单元", "圆" @@ -61368,7 +61493,9 @@ "002279": { "id": "002279", "content": "过点$P(1,2)$的直线$l$将圆$x^2+y^2-4x-5=0$分成两个弓形.\\\\ \n(1) 当大小两个弓形的面积之差最大时, 直线$l$的方程为\\blank{100};\\\\ \n(2) (选做)说明第(1)题的理由.", - "objs": [], + "objs": [ + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -61393,7 +61520,9 @@ "002280": { "id": "002280", "content": "两圆$x^2+y^2-10x-10y=0$与$x^2+y^2+6x+2y-40=0$的公共弦所在的直线方程为\\blank{100}, 公共弦长为\\blank{50}.", - "objs": [], + "objs": [ + "K0712004X" + ], "tags": [ "第七单元", "圆" @@ -61418,7 +61547,9 @@ "002281": { "id": "002281", "content": "过圆$x^2+y^2+2x-4y-5=0$和直线$2x+y+4=0$的两个交点, 且面积最小的圆的一般方程是\\blank{50}.", - "objs": [], + "objs": [ + "K0712004X" + ], "tags": [ "第七单元", "圆" @@ -61443,7 +61574,10 @@ "002282": { "id": "002282", "content": "两圆$x^2+y^2+2ax+2ay+2a^2-1=0$与$x^2+y^2+2bx+2by+2b^2-2=0$的公共弦长的最大值是\\blank{50}.", - "objs": [], + "objs": [ + "K0712004X", + "K0710002X" + ], "tags": [ "第七单元", "圆" @@ -61468,7 +61602,9 @@ "002283": { "id": "002283", "content": "已知两圆$x^2+y^2+4x-4y-1=0$与$x^2+y^2+2x+2y-2=0$相交于$P,Q$两点, 则公共弦$PQ$的中垂线的方程是\\blank{100}.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -61493,7 +61629,11 @@ "002284": { "id": "002284", "content": "过两圆$x^2+y^2+4x-3=0$与$x^2+y^2-4y-3=0$的交点, 且圆心在直线$2x-y-4=0$上的圆的一般方程是\\blank{100}.", - "objs": [], + "objs": [ + "K0712004X", + "K0710004X", + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -61518,7 +61658,11 @@ "002285": { "id": "002285", "content": "过点$P(-2,-3)$作圆$C:{{(x-4)}^{2}}+{{(y-2)}^{2}}=9$的两条切线, 切点分别为$A,B$, 求:\\\\ \n(1) 过圆心$C$, 切点$A,B$这三点的圆的方程;\\\\ \n(2) 直线$AB$的方程;\\\\ \n(3) 线段$AB$的长.", - "objs": [], + "objs": [ + "K0709003X", + "K0712004X", + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -61543,7 +61687,9 @@ "002286": { "id": "002286", "content": "已知两圆的方程为$x^2+y^2=1$与$(x-3)^2+(y-6)^2=4$.\\\\ \n(1) 求出两条外公切线的交点$A$及两条内公切线的交点$B$的坐标;\\\\ \n(2) 求四条公切线的方程.", - "objs": [], + "objs": [ + "K0711004X" + ], "tags": [ "第七单元", "圆" @@ -61568,7 +61714,9 @@ "002287": { "id": "002287", "content": "[选做]\n设外切两圆的方程分别为${{x}^{2}}+{{y}^{2}}+{{D}_{1}}x+{{E}_{1}}y+{{F}_{1}}=0$与${{x}^{2}}+{{y}^{2}}+{{D}_{2}}x+{{E}_{2}}y+{{F}_{2}}=0$.\n求证: 内公切线的方程为${{D}_{1}}x+{{E}_{1}}y+{{F}_{1}}={{D}_{2}}x+{{E}_{2}}y+{{F}_{2}}$.", - "objs": [], + "objs": [ + "K0712004X" + ], "tags": [ "第七单元", "圆" @@ -61593,7 +61741,9 @@ "002288": { "id": "002288", "content": "[选做]\n已知三圆$C_1,C_2,C_3$两两相交, 它们两两之间的公共弦所在的直线分别记为$l_{12}$, $l_{23}$, $l_{31}$, 若$l_{12},l_{23}$均经过点$P$, 证明: $l_{13}$也经过点$P$.", - "objs": [], + "objs": [ + "K0712004X" + ], "tags": [ "第七单元", "圆" @@ -61618,7 +61768,9 @@ "002289": { "id": "002289", "content": "已知实数$a,b,c$满足$3(a^2+b^2)=4c^2(c\\ne 0)$, 则直线$ax+by+c=0$与圆$x^2+y^2=1$的关系是\\bracket{20}.\n\\fourch{相交}{相切}{相离}{不确定}", - "objs": [], + "objs": [ + "K0711001X" + ], "tags": [ "第七单元", "圆" @@ -61643,7 +61795,9 @@ "002290": { "id": "002290", "content": "圆$x^2+y^2-2x=3$与直线$y=ax+1$的交点个数是\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{随$a$的不同而改变}%2 C", - "objs": [], + "objs": [ + "K0711001X" + ], "tags": [ "第七单元", "圆" @@ -61668,7 +61822,9 @@ "002291": { "id": "002291", "content": "已知圆$x^2+y^2=4$上有一点$P$, 它到直线$4x+3y=2$的距离取到最大值, 则$P$的坐标为\\blank{50}.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -61768,7 +61924,9 @@ "002295": { "id": "002295", "content": "已知定圆$x^2+y^2=8$和定点$P(4,0)$, 过$P$点作直线$l$, 若这条直线$l$与已知圆相交, 则直线$l$的倾斜角的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -62015,7 +62173,9 @@ "002306": { "id": "002306", "content": "写出分别满足下列条件的椭圆的标准方程.\\\\ \n(1) 焦点坐标$(6,0),(-6,0)$, 且椭圆过$(0,8)$.\\blank{100}\\\\ \n(2) 焦距为$12$, 且椭圆过$(0,8)$.\\blank{100}\\\\ \n(3) 椭圆过点$(0,-2),(1,0)$.\\blank{100}", - "objs": [], + "objs": [ + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -62037,7 +62197,10 @@ "002307": { "id": "002307", "content": "椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$的焦点坐标为\\blank{50}, 离心率为\\blank{50}, 准线方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X", + "K0714005X" + ], "tags": [ "第七单元", "椭圆" @@ -62061,7 +62224,10 @@ "002308": { "id": "002308", "content": "椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{25}=1$的焦点坐标为\\blank{50}, 离心率为\\blank{50}, 准线方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X", + "K0714005X" + ], "tags": [ "第七单元", "椭圆" @@ -62085,7 +62251,9 @@ "002309": { "id": "002309", "content": "已知$F_1,F_2$是椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\ (a>b>0)$的两个焦点, $AB$是过$F_1$的弦, 则$\\triangle ABF_2$的周长为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -62107,7 +62275,9 @@ "002310": { "id": "002310", "content": "若方程$\\dfrac{x^2}{25-m}+\\dfrac{y^2}{16+m}=1$表示焦点在$y$轴上的椭圆, 则实数$m$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -62131,7 +62301,9 @@ "002311": { "id": "002311", "content": "求过点$(-\\dfrac{3}{2},\\dfrac{5}{2})$与$(\\sqrt 3,\\sqrt 5)$的椭圆的标准方程.", - "objs": [], + "objs": [ + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -62176,7 +62348,9 @@ "002313": { "id": "002313", "content": "[选做]\n平面上有一定直线$l$和$l$外一定点$F$. 求证: 当一个动点$P$到$F$的距离和它到$l$的距离之比是一个小于$1$的常数时, 点$P$的轨迹是椭圆.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -62198,7 +62372,9 @@ "002314": { "id": "002314", "content": "若方程$\\dfrac{x^2}{k-5}+\\dfrac{y^2}{3-k}=-1$表示椭圆, 则实数$k$的取值范围为\\blank{80}.\n%\\ans{$(3,4)\\cup (4,5)$}", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -62220,7 +62396,9 @@ "002315": { "id": "002315", "content": "已知椭圆$mx^2+y^2=9$与椭圆$9x^2+25y^2=100$的焦距相等, 则实数$m=$\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -62242,7 +62420,9 @@ "002316": { "id": "002316", "content": "已知$bb>0)$上的任一点, $F_1,F_2$是它的左, 右焦点, 则$|PF_1|\\times |PF_2|$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -62395,7 +62585,9 @@ "002323": { "id": "002323", "content": "已知$F_1$是椭圆$5x^2+9y^2=45$的左焦点, $P$是此椭圆上的动点, $A(1,1)$是一定点, 则$|PA|+|PF_1|$的最大值为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -62417,7 +62609,9 @@ "002324": { "id": "002324", "content": "已知点$P$在椭圆$\\dfrac{x^2}{100}+\\dfrac{y^2}{36}=1$上, 它到椭圆左焦点$F_1$的距离是\n它到椭圆右焦点$F_2$的距离的$3$倍.\\\\ \n(1) 求$|PF_1|,|PF_2|$;\\\\ \n(2) 求点$P$的坐标.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -62439,7 +62633,9 @@ "002325": { "id": "002325", "content": "已知椭圆$\\dfrac{x^2}{2}+y^2=1$,\n直线$l$交椭圆于$A,B$两点, 若线段$AB$的中点坐标为$M(\\dfrac{1}{2},\\dfrac{1}{2})$, 求直线$l$的方程.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -62461,7 +62657,9 @@ "002326": { "id": "002326", "content": "已知椭圆$C: \\dfrac{x^2}{9}+y^2=1$, $P$是曲线上的动点, 定点$A$的坐标为$(m,0)$, 其中$m$是实常数.\n求$|PA|$的最小值.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -62483,7 +62681,10 @@ "002327": { "id": "002327", "content": "一个焦点把长轴分成长度为$7$和$1$两段的椭圆的标准方程为\\blank{100}.\n%\\ans{$x^2/16+y^2/7=1$or$x^2/7+y^2/16=1$}", - "objs": [], + "objs": [ + "K0714003X", + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -62505,7 +62706,9 @@ "002328": { "id": "002328", "content": "已知长轴长与短轴长之比为$2:1$, 一条准线方程为$x+4=0$的椭圆的标准方程为\\blank{100}.\n%\\ans{$\\dfrac{x^2}{12}+\\dfrac{y^2}{3}=1$}", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第七单元", "椭圆" @@ -62527,7 +62730,9 @@ "002329": { "id": "002329", "content": "以直线$3x+4y-12=0$和两轴的交点之一作为顶点, 另一交点作为焦点的椭圆的标准方程为\\blank{100}.\n%\\ans{$x^2/25+y^2/9=1$or$x^2/16+y^2/25=1$}", - "objs": [], + "objs": [ + "K0713003X" + ], "tags": [ "第七单元", "椭圆" @@ -62549,7 +62754,10 @@ "002330": { "id": "002330", "content": "过点$P(3,0)$, 且长轴长是短轴长的三倍的椭圆的标准方程为\\blank{100}.\n%\\ans{$x^2/9+y^2=1$or$x^2/9+y^2/81=1$}", - "objs": [], + "objs": [ + "K0714003X", + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -62571,7 +62779,9 @@ "002331": { "id": "002331", "content": "已知$M$为椭圆上一点, $F_1,F_2$是两个焦点, 且$\\angle MF_1F_2=2\\alpha$, $\\angle MF_2F_1=\\alpha$, 则椭圆的离心率为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -62593,7 +62803,9 @@ "002332": { "id": "002332", "content": "已知$P$是椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$上的点, 且$\\angle F_1PF_2=90^\\circ$($F_1,F_2$是该椭圆的两个焦点), 则$\\triangle F_1PF_2$的面积为\\blank{50}, $P$的坐标为\\blank{50}.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -62615,7 +62827,9 @@ "002333": { "id": "002333", "content": "已知$F$是椭圆$b^2x^2+a^2y^2=a^2b^2 \\ (a>b>0)$的一个焦点, $PQ$是过其中心的一条弦, 则$\\triangle PQF$面积的最大值是\\blank{50}.", - "objs": [], + "objs": [ + "K0714002X" + ], "tags": [ "第七单元", "椭圆" @@ -62637,7 +62851,9 @@ "002334": { "id": "002334", "content": "已知直线$l:y=kx+1$, 若不论$k$取何值, $l$总与椭圆$\\dfrac{x^2}{5}+\\dfrac{y^2}{m}=1$总有公共点,\n则常数$m$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0713001X" + ], "tags": [ "第七单元", "椭圆" @@ -62659,7 +62875,10 @@ "002335": { "id": "002335", "content": "以椭圆的两个焦点为直径端点的圆交椭圆于四个点, 若顺次连接这四个点及两个焦点恰好组成一个正六边形, 则椭圆的离心率为\\blank{50}.\n%\\ans{$\\sqrt{3}-1$}", - "objs": [], + "objs": [ + "K0714005X", + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -62702,7 +62921,9 @@ "002337": { "id": "002337", "content": "已知椭圆$x^2+2y^2=12$及$x$轴正向上一定点$A$, 过$A$作斜率为$1$的直线, 此直线被椭圆截得的弦长为$\\dfrac{4\\sqrt{14}}{3}$, 求$A$的坐标.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -62724,7 +62945,9 @@ "002338": { "id": "002338", "content": "已知$P$是椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$上的点, 且$\\angle F_1PF_2=\\theta$($F_1,F_2$是该椭圆的两个焦点), 试用$a,b,\\theta$表示$\\triangle F_1PF_2$的面积.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -62746,7 +62969,9 @@ "002339": { "id": "002339", "content": "已知椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$与直线$x+2y-2=0$交于$A,B$两点, $|AB|=5$,\n且$AB$中点的坐标为$(m,\\dfrac{1}{2})$, 求此椭圆的方程. (提示: 算法合适的话, 此题不用联立椭圆与直线方程. )", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -62816,7 +63041,10 @@ "002342": { "id": "002342", "content": "以椭圆的右焦点$F_2$为圆心作圆, 使这个圆通过椭圆的中心, 且交椭圆于$M$点, 若直线$MF_1$($F_1$为左焦点)是圆$F_2$的切线, 则椭圆的离心率为\\blank{50}.", - "objs": [], + "objs": [ + "K0714005X", + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -62838,7 +63066,10 @@ "002343": { "id": "002343", "content": "已知圆柱底面的直径为$2R$, 一个与底面成$30^\\circ$角的平面截这个圆柱, 截得的曲线是椭圆.\n这个椭圆的离心率为\\blank{50}.", - "objs": [], + "objs": [ + "K0714005X", + "K0714002X" + ], "tags": [ "第七单元", "椭圆" @@ -62860,7 +63091,9 @@ "002344": { "id": "002344", "content": "已知椭圆的中心在原点, 长轴在$x$轴上, 直线$x+y=1$被椭圆截得的弦$AB$长为$2\\sqrt{2}$, 且$AB$的中点与椭圆中心连线的斜率为$\\dfrac{\\sqrt{2}}{2}$, 则这个椭圆的方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -62882,7 +63115,9 @@ "002345": { "id": "002345", "content": "已知点$P$在圆$x^2+(y-4)^2=1$上移动, 点$Q$在椭圆$\\dfrac{x^2}{4}+y^2=1$上移动, 则$|PQ|$的最大值为\\blank{50}.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -62904,7 +63139,9 @@ "002346": { "id": "002346", "content": "已知$\\triangle ABC$的三个顶点均在椭圆$4x^2+5y^2=80$上, 且点$A$是椭圆短轴的下端点, $\\triangle ABC$的重心是椭圆的右焦点, 求直线$BC$的方程.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -62926,7 +63163,9 @@ "002347": { "id": "002347", "content": "已知椭圆$\\dfrac{x^2}{2}+y^2=1$.\\\\ \n(1) 求斜率为$2$的平行弦的中点轨迹方程;\\\\ \n(2) 过$A(2,1)$引椭圆的割线, 若截得的弦的中点落在一条二次曲线上, 求这个二次曲线的方程, 并回答(只需给出答案)中点是否能取遍该二次曲线的每一点?", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -63127,7 +63366,9 @@ "002356": { "id": "002356", "content": "已知$P,Q$分别是椭圆$9x^2+4y^2=36$的两个焦点, $M$在双曲线$9x^2-25y^2=225$上, 则$\\triangle PQM$重心的轨迹方程为\\blank{80}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆", @@ -63279,7 +63520,9 @@ "002363": { "id": "002363", "content": "与两圆$x^2+y^2=1$和$x^2+y^2-8x+7=0$都相切的圆的圆心轨迹是\\bracket{20}.\n\\twoch{两个椭圆}{两条双曲线}{一条双曲线和一条直线}{一个椭圆和一条双曲线}", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆", @@ -63368,7 +63611,9 @@ "002367": { "id": "002367", "content": "已知椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1\\ (a>b>0)$和双曲线$\\dfrac{x^2}{m^2}-\\dfrac{y^2}{n^2}=1 \\ (m,n>0)$有公共的焦点$F_1,F_2$, $P$是两曲线的一个交点.\\\\ \n(1) 证明: $\\angle F_1PF_2=2\\arctan \\dfrac{n}{b}$;\\\\ \n(2) 证明: $\\triangle F_1PF_2$的面积为$bn$.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆", @@ -63660,7 +63905,9 @@ "002380": { "id": "002380", "content": "与椭圆$x^2+4y^2=64$有共同焦点, 且一条渐近线的方程为$x+\\sqrt{3}y=0$的双曲线的标准方程为\\blank{80}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆", @@ -63857,7 +64104,9 @@ "002389": { "id": "002389", "content": "已知椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$上三点$A(x_1,y_1)$, $B(4,y_2)$, $C(x_3,y_3)$和焦点$F(4,0)$的距离依次成等差数列.\\\\ \n(1) 求$x_1+x_3$;\\\\ \n(2) 证明: 线段$AC$的垂直平分线过定点, 并求出此定点的坐标.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -64727,7 +64976,10 @@ "002428": { "id": "002428", "content": "已知抛物线$y^2=4x$与椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{m}=1$有共同的焦点$F_2$.\\\\ \n(1) 求$m$的值;\\\\ \n(2) 若$P$是两曲线的一个公共点, $F_1$是椭圆的另一个焦点, 且$\\angle PF_1F_2=\\alpha$, $\\angle PF_2F_1=\\beta$, 求$\\cos\\alpha\\cos\\beta$.\\\\ \n(3) 求$\\triangle PF_1F_2$的面积.", - "objs": [], + "objs": [ + "K0713002X", + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -65105,7 +65357,9 @@ "002443": { "id": "002443", "content": "以$F_1(0,-1)$, $F_2(0,3)$为两个焦点, 又过点$A(2,1)$的椭圆的方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -65203,7 +65457,9 @@ "002447": { "id": "002447", "content": "椭圆$\\dfrac{(x-1)^2}{16}+\\dfrac{(y-2)^2}{9}=1$关于点$M(2,-1)$对称的椭圆的方程为\\blank{50}.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第七单元", "椭圆" @@ -65754,7 +66010,9 @@ "002469": { "id": "002469", "content": "椭圆$\\left\\{\\begin{array}{l}x=4+2\\cos\\theta,\\\\y=1+5\\sin\\theta,\\end{array}\\right.$的焦点坐标为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -66027,7 +66285,9 @@ "002480": { "id": "002480", "content": "已知直线$l$过点$P(0,3)$, 倾斜角为$\\alpha$, 且与椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$交\n于$A,B$两点(可重合), 求$|PA|\\cdot|PB|$的最大值.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -66052,7 +66312,9 @@ "002481": { "id": "002481", "content": "如图, $AB,CD$是椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$的两条相交弦, 交点为$P$. 两弦与椭圆长轴的夹角均为$\\alpha$. 求证: $A,B,C,D$四点共圆.\n\\begin{center}\n\\begin{tikzpicture}[>=latex][scale = 1.5]\n \\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n \\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,0) ellipse (2 and 1);\n \\draw (-1.3271,0.7481) node [above left] {$D$} -- (1.9447,-0.2334) node [below right] {$B$};\n \\draw (-1.7575,-0.4773) node [below left] {$C$} -- (1.6693,0.5508) node [above right] {$A$};\n \\draw (0.5,0.2) node [above] {$P$};\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -66077,7 +66339,9 @@ "002482": { "id": "002482", "content": "[选做]\n已知$P(1,1)$是椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{4}=1$的弦$AB$的一个三等分点,\n求弦$AB$所在直线的方程.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -71116,7 +71380,9 @@ "002683": { "id": "002683", "content": "设椭圆$E$的方程为$\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$, $F(-1,0)$是椭圆的左焦点, $P$是椭圆$E$上的一个动点, $A(1,1)$是椭圆内一点.\\\\ \n(1) 求$|PA|+2|PF|$的最小值;\\\\ \n(2) 求$|PA|+|PF|$的最小值及最大值.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -71182,7 +71448,9 @@ "002686": { "id": "002686", "content": "已知方程$7x^2-2xy+7y^2=48$表示一个椭圆, 求它的任意一组焦点和准线.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第七单元", "椭圆" @@ -71226,7 +71494,9 @@ "002688": { "id": "002688", "content": "设椭圆$E$的方程为$x^2+\\dfrac{y^2}{4}=1$. $AB$是椭圆$E$的一条动弦, 其中点记为$M$.\\\\ \n(1) 若$|AB|=2$, 求$M$的纵坐标的最小值; $-\\dfrac{2\\sqrt{3}}{3}$\\\\ \n(2) 若$|AB|=\\dfrac{1}{2}$, 求$M$的纵坐标的最小值. $-\\dfrac{\\sqrt{15}}{2}$", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -88758,7 +89028,9 @@ "003381": { "id": "003381", "content": "已知三点$A(2,-3)$、$B(-2,-5)$, 求分别满足下列条件的圆方程:\\\\\n(1) 以$A$、$B$两点为直径的圆为\\blank{50};\\\\\n(2) 过$A$、$B$两点, 且圆心在直线$x-2y-3=0$上的圆为\\blank{50}.", - "objs": [], + "objs": [ + "K0710005X" + ], "tags": [ "第七单元", "圆" @@ -88780,7 +89052,9 @@ "003382": { "id": "003382", "content": "若方程$x^2+y^2+ax+2ay+2a^2+a-1=0$表示圆, 则$a$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0710002X" + ], "tags": [ "第七单元", "圆" @@ -88804,7 +89078,10 @@ "003383": { "id": "003383", "content": "(1) 过点$(3,4)$作圆$x^2+y^2=25$的切线$l$, 则$l$的方程为\\blank{50};\\\\\n(2) 过点$(2,4)$作圆$x^2+y^2-2x=0$的切线$m$, 则$m$的方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0711003X", + "K0711004X" + ], "tags": [ "第七单元", "圆" @@ -88826,7 +89103,9 @@ "003384": { "id": "003384", "content": "若点$P$在圆$x^2+y^2+4x-6y+12=0$上运动, 点$Q$在直线$4x+3y=21$上运动, 则$|PQ|$的最小值是\\blank{50}.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -88850,7 +89129,9 @@ "003385": { "id": "003385", "content": "已知圆$x^2+y^2=8$内一点$P(-1,2)$, 过点$P$作直线$l$交圆于$A$、$B$, 若弦$AB$恰被点$P$平分, 则直线$l$的方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0710006X" + ], "tags": [ "第七单元", "圆" @@ -88872,7 +89153,10 @@ "003386": { "id": "003386", "content": "若直线$ax+by=1$与圆$C:x^2+y^2=1$相交, 则点$P(a,b)$与圆$C$的位置关系是\\bracket{20}.\n\\twoch{点在圆内}{点在圆上}{点在圆外}{随$a,b$取值的变化而变化}", - "objs": [], + "objs": [ + "K0711002X", + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -88941,7 +89225,11 @@ "003389": { "id": "003389", "content": "已知圆$C:x^2+y^2-4x-14y+45=0$及点$Q(-2,3)$.\\\\\n(1) 若$P(m,m+1)$在圆$C$上, 求线段$PQ$的长及直线$PQ$的斜率;\\\\\n(2) 若$P$为圆$C$上任意一点, 求线段$PQ$的长的最大值和最小值;\\\\\n(3) 若点$M(a,b)$在圆$C$上, 求$u=\\dfrac{b-3}{a+2}$的最大值和最小值.", - "objs": [], + "objs": [ + "K0709001X", + "K0712002X", + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -88963,7 +89251,9 @@ "003390": { "id": "003390", "content": "过原点的直线与圆$x^2+y^2-6x+5=0$相交于$A$、$B$两点, 求弦$AB$的中点$M$的轨迹方程.", - "objs": [], + "objs": [ + "K0711005X" + ], "tags": [ "第七单元", "圆" @@ -88985,7 +89275,9 @@ "003391": { "id": "003391", "content": "已知圆$C:x^2+y^2=25$, 过定点$P(-4,0)$的直线$l$交圆$C$于$A$、$B$两点.\\\\\n(1) 若直线$l$的斜率为$1$, 求弦长$|AB|$;\\\\\n(2) 求弦长$|AB|$的取值范围;\\\\\n(3) 求$\\triangle AOB$面积的取值范围.", - "objs": [], + "objs": [ + "K0710006X" + ], "tags": [ "第七单元", "圆" @@ -89007,7 +89299,9 @@ "003392": { "id": "003392", "content": "圆$x^2+y^2-2x-4y-1=0$关于直线$x-y+3=0$的对称的曲线的方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -89029,7 +89323,9 @@ "003393": { "id": "003393", "content": "若直线$mx+ny-3=0$与圆$x^2+y^2=3$没有公共点, 则$m^2+n^2$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -89051,7 +89347,9 @@ "003394": { "id": "003394", "content": "方程$|x|-1=\\sqrt{1-y^2}$表示的曲线是\\bracket{20}.\n\\twoch{一条直线}{两条射线}{一个圆}{两个半圆(即两段圆弧)}", - "objs": [], + "objs": [ + "K0709001X" + ], "tags": [ "第七单元", "圆" @@ -89073,7 +89371,10 @@ "003395": { "id": "003395", "content": "求满足下列条件的圆的方程:\\\\\n(1) 经过点$(2,-1)$且和直线$x-y=1$相切, 同时圆心在直线$y=-2x$上的圆的方程为\\blank{50};\\\\\n(2) 经过点$A(-2,-4)$, 且与直线$l:x+3y-26=0$相切于点$B(8,6)$的圆的方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0709007X", + "K0710005X" + ], "tags": [ "第七单元", "圆" @@ -89095,7 +89396,9 @@ "003396": { "id": "003396", "content": "已知方程$x^2+y^2+Dx+Ey+F=0$表示一个圆. ``$D^2=4F$''是``该圆与$x$轴相切''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分又非必要}", - "objs": [], + "objs": [ + "K0710002X" + ], "tags": [ "第七单元", "圆" @@ -89139,7 +89442,9 @@ "003398": { "id": "003398", "content": "已知$m\\in \\mathbf{R}$, 直线$l:mx-(m^2+1)y=4m$和圆$C:x^2+y^2-8x+4y+16=0$.\\\\\n(1) 求直线$l$的斜率$k$的取值范围;\\\\\n(2) 直线$l$能否将圆$C$分割成弧长的比值为$\\dfrac{1}{2}$的两段圆弧? 为什么?", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -89161,7 +89466,9 @@ "003399": { "id": "003399", "content": "若点$P(-3,0)$是椭圆$x^2+2y^2-k=0$上的点, 则椭圆的焦点坐标是\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -89183,7 +89490,9 @@ "003400": { "id": "003400", "content": "方程$\\dfrac{x^2}{k-5}+\\dfrac{y^2}{3-k}=-1$表示焦点在$y$轴上的椭圆, 则实数$k$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -89205,7 +89514,10 @@ "003401": { "id": "003401", "content": "(1) 焦距是$2\\sqrt 5$, 长轴长是$8$的椭圆的标准方程是\\blank{50};\\\\\n(2) 长轴长是短轴长的2倍, 且经过点$(2,1)$的椭圆的标准方程是\\blank{50};\\\\\n(3) 经过点$A(\\sqrt 3,-2)$、$B(\\sqrt 5,\\dfrac{\\sqrt{30}}3)$的椭圆的方程是\\blank{50}.", - "objs": [], + "objs": [ + "K0713003X", + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -89227,7 +89539,9 @@ "003402": { "id": "003402", "content": "已知点$P$是椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$上一点, $F_1F_2$是焦点, 若$\\angle F_1PF_2=60^\\circ$, 则三角形$F_1PF_2$的面积为\\blank{50}.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -89249,7 +89563,9 @@ "003403": { "id": "003403", "content": "已知椭圆$\\dfrac{x^2}{36}+\\dfrac{y^2}{16}$=1的弦过点$P(3,2)$且被$P$平分, 则此弦所在的直线方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -89271,7 +89587,9 @@ "003404": { "id": "003404", "content": "椭圆$\\dfrac{x^2}{45}+\\dfrac{y^2}{20}=1$的焦点为$F_1$、$F_2$, 过原点$O$作直线交椭圆于$A$、$B$两点, 若$\\triangle ABF_2$的面积为$20$, 则点$A$的纵坐标为\\blank{50}.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -89314,7 +89632,9 @@ "003406": { "id": "003406", "content": "已知椭圆$\\dfrac{x^2}m+\\dfrac{y^2}6=1$, $F_1,F_2$是它的两个焦点, 若椭圆上存在两个不同的点$P$, 使$\\angle F_1PF_2=90^\\circ$, 则$m=$\\blank{50}.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -89336,7 +89656,9 @@ "003407": { "id": "003407", "content": "已知椭圆$\\dfrac{y^2}9+{x^2}=1$, 一条不与坐标轴平行的直线$l$与该椭圆交于不同的两点$M$、$N$, 且线段$MN$的中点的横坐标为$-\\dfrac 12$.\\\\\n(1) 求直线$l$的斜率的取值范围;\\\\\n(2) 求直线$l$的倾斜角的取值范围.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -89358,7 +89680,9 @@ "003408": { "id": "003408", "content": "已知椭圆$\\dfrac{x^2}2+{y^2}=1$.\\\\\n(1) *过椭圆的左焦点$F$引椭圆的割线, 求截得的弦的中点$P$的轨迹方程;\\\\\n(2) 求斜率为2的平行弦中点$Q$的轨迹方程;\\\\\n(3) 求过点$M(\\dfrac 12,\\dfrac 12)$且被$M$平分的弦所在直线方程.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -89380,7 +89704,10 @@ "003409": { "id": "003409", "content": "设椭圆$C:\\dfrac{x^2}2+{y^2}=1$的右焦点为$F$, 过$F$的直线$l$与$C$交于$AB$两点, 点$M$的坐标为$(2,0)$.\\\\\n(1) 当$l$与$x$轴垂直时, 求直线$AM$的方程;\\\\\n(2) 设$O$为坐标原点, 证明: $\\angle OMA=\\angle OMB$.", - "objs": [], + "objs": [ + "K0713002X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -89402,7 +89729,9 @@ "003410": { "id": "003410", "content": "若椭圆的中心为原点, 焦点在坐标轴上, 焦点到长轴端点的距离分别为$\\sqrt 2-1$与$\\sqrt 2+1$, 则椭圆的方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0713003X" + ], "tags": [ "第七单元", "椭圆" @@ -89424,7 +89753,10 @@ "003411": { "id": "003411", "content": "与椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$有相同的焦点, 且经过点$(3,-2)$的椭圆为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X", + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -89468,7 +89800,9 @@ "003413": { "id": "003413", "content": "椭圆${x^2}+\\dfrac{y^2}4=1$上的点$P(x,y)$到定直线$x+y-6=0$的最远距离是\\blank{50}.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -89490,7 +89824,9 @@ "003414": { "id": "003414", "content": "记椭圆$\\dfrac{x^2}4+\\dfrac{n{y^2}}{4n+1}=1$围成的区域(含边界)为$\\Omega_n \\ (n=1,2,\\cdots)$, 当点$(x,y)$分别在$\\Omega_1,\\Omega_2,\\cdots$上时, $x+y$的最大值分别是$M_1,M_2,\\cdots$, 则$\\displaystyle\\lim_{n\\to \\infty}M_n=$\\blank{50}.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第七单元", "椭圆" @@ -89512,7 +89848,9 @@ "003415": { "id": "003415", "content": "椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$上的动点$P(x,y)$与定点$M(m,0)$($0n>0$)和双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)有相同的焦点$F_1,F_2$, 点$P$是椭圆和双曲线的一个交点.\\\\\n(1) 求证: $|PF_1|\\cdot |PF_2|=m^2-a^2$;\\\\\n(2) 求证: $\\triangle PF_1F_2$的面积$S=nb$.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆", @@ -93797,7 +94139,9 @@ "003591": { "id": "003591", "content": "已知圆$x^2+y^2-2x-4y=0$, 则该圆的圆心坐标为\\blank{50}.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -94211,7 +94555,9 @@ "003608": { "id": "003608", "content": "已知椭圆$\\Gamma:\\dfrac{x^2}2+y^2=1$, $F_1, F_2$是其左右焦点, 直线$l$过点$P(m,0) \\ (m<-\\sqrt2)$, 交椭圆$\\Gamma$于$A,B$两点, 且$A,B$都在$x$轴上方, 点$A$在线段$BP$上.\\\\\n(1) 若$B$是上顶点, $|\\overrightarrow{BF_1}|=|\\overrightarrow{PF_1}|$, 求$m$的值;\\\\\n(2) 若$\\overrightarrow{F_1A}\\cdot \\overrightarrow{F_2A}=\\dfrac13$, 且原点$O$到直线$l$的距离为$\\dfrac{4\\sqrt{15}}{15}$, 求直线$l$的方程;\\\\\n(3) 对于任意点$P$, 是否存在唯一直线$l$, 使得$\\overrightarrow{F_1A}\\parallel \\overrightarrow{F_2B}$成立? 若存在, 求出直线$l$的斜率; 若不存在, 请说明理由.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -94486,7 +94832,9 @@ "003619": { "id": "003619", "content": "已知椭圆$C:\\dfrac{x^2}4+\\dfrac{y^2}3=1$, 直线$l$经过椭圆右焦点$F$, 交椭圆$C$于$P,Q$两点(点$P$在第二象限), 若$Q$关于$x$轴对称的点为$Q'$, 且满足$PQ\\perp FQ'$, 则直线$l$的方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -95271,7 +95619,9 @@ "003650": { "id": "003650", "content": "已知椭圆$\\dfrac{x^2}{8}+\\dfrac{y^2}{4}=1$, $F_1$、$F_2$为左、右焦点, 直线$l$过$F_2$, 交椭圆于$A$、$B$两点.\\\\\n(1) 若直线$l$垂直于$x$轴, 求$|AB|$;\\\\\n(2) 当$\\angle F_1AB=90^\\circ$, $A$在$x$轴上方时, 求$A$、$B$的坐标;\\\\\n(3) 若直线$AF_1$交$y$轴于$M$, 直线$BF_1$交$y$轴于$N$, 是否存在直线$l$, 使得$S_{\\triangle F_1AB}=S_{\\triangle F_1MN}$? 若存在, 求出直线$l$的方程; 若不存在, 说明理由.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -95625,7 +95975,9 @@ "003664": { "id": "003664", "content": "设$P$是椭圆$\\dfrac{x^2}{5}+\\dfrac{y^2}{3}=1$上的动点, 则$P$到该椭圆的两个焦点的距离之和为\\bracket{15}.\n\\fourch{$2\\sqrt{2}$}{$2\\sqrt{3}$}{$2\\sqrt{5}$}{$4\\sqrt{2}$}", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -96620,7 +96972,9 @@ "003704": { "id": "003704", "content": "已知过原点$O$的直线与椭圆$C:\\dfrac{x^2}4+{y^2}=1$交于$A,B$两点, 点$A$到$y$轴的距离$d$满足$d\\in [1,2)$, 点$D$在椭圆$C$上, 且$AD\\perp AB$, 直线$BD$与$x$轴、$y$轴分别交于$M,N$两点.\\\\\n(1) 设直线$BD,AM$的斜率分别为$k_1,k_2$, 求$k_1\\cdot k_2$的取值范围;\\\\\n(2) 求$\\triangle OMN$面积的最大值.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -97298,7 +97652,9 @@ "003733": { "id": "003733", "content": "已知椭圆中心在原点, 一个焦点为$F(-2\\sqrt{3},0)$, 且长轴长是短轴长的$2$倍, 则该椭圆的标准方程是\\blank{50}.", - "objs": [], + "objs": [ + "K0713003X" + ], "tags": [ "第七单元", "椭圆" @@ -97641,7 +97997,9 @@ "003748": { "id": "003748", "content": "(理科)设曲线$C$定义为到点$(-1,-1)$和$(1,1)$距离之和为$4$的动点的轨迹. 若将曲线$C$绕坐标原点逆时针旋转$45^\\circ$, 则此时曲线$C$的方程为\\blank{50}.\\\\\n(文科)椭圆$2x^2+3y^2=6$的焦距为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第三单元", "第七单元", @@ -99435,7 +99793,9 @@ "003826": { "id": "003826", "content": "已知$F_1,F_2$为椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$的左、右焦点, $P$为椭圆上一点, $M$是$F_1P$的中点, $|OM|=3$, 则点$M$到椭圆左焦点的距离为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -100144,7 +100504,9 @@ "003858": { "id": "003858", "content": "椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\ (a>b>0)$和圆$x^2+y^2=\\left(\\dfrac b2+c\\right)^2 \\ (c^2=a^2-b^2)$有两个不同的公共点, 则$\\dfrac ca$的值是\\blank{50}.", - "objs": [], + "objs": [ + "K0714004X" + ], "tags": [ "第七单元", "椭圆" @@ -100579,7 +100941,9 @@ "003877": { "id": "003877", "content": "已知点$P(x,y)$是直线$kx+y+4=0 \\ (k>0)$上一动点, $PA,PB$是圆$C:x^2+y^2-2y=0$的两条切线, $A,B$是切点, 若四边形$PACB$($C$为圆心)面积的最小值为$2$, 则$k$的值为\\bracket{20}.\n\\fourch{$2$}{$\\dfrac{\\sqrt{21}}{2}$}{$2\\sqrt{2}$}{$3$}", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -100941,7 +101305,10 @@ "003893": { "id": "003893", "content": "已知圆方程为$x^2+y^2-2ax-4ay+4a^2+t=0 \\ (a\\ne 0)$.\\\\\n(1) 若$t=\\dfrac 12 a^2$, 确定无论$a$为何值均与圆相切的直线的方程;\\\\\n(2) 若$t=a^2-4$, 确定无论$a$为何值被圆截得的弦长为$1$的直线的方程.", - "objs": [], + "objs": [ + "K0710003X", + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -100986,7 +101353,9 @@ "003895": { "id": "003895", "content": "已知椭圆$\\dfrac{x^2}{t^2}+\\dfrac{y^2}{5t}=1$的焦距为$2\\sqrt{6}$, 则实数$t=$\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -101303,7 +101672,10 @@ "003909": { "id": "003909", "content": "已知椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\ (a>b>0)$的一个焦点坐标为$(1,0)$, 且长轴长是短轴长的$\\sqrt{2}$倍.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 设$O$为坐标原点, 椭圆$C$与直线$y=kx+1$相交于两个不同的点$A,B$, 线段$AB$的中点为$P$, 若直线$OP$的斜率为$-1$, 求$\\triangle AOB$的面积.", - "objs": [], + "objs": [ + "K0713003X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -101503,7 +101875,9 @@ "003918": { "id": "003918", "content": "椭圆两焦点为$F_1(-4,0)$, $F_2(4,0)$, $P$在椭圆上, 若$\\triangle PF_1F_2$的面积的最大值为$12$, 则该椭圆的标准方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0713003X" + ], "tags": [ "第七单元", "椭圆" @@ -105285,7 +105659,9 @@ "004078": { "id": "004078", "content": "(1) 设椭圆$C_1:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$与双曲线$C_2:9{x^2}-\\dfrac{9y^2}8=1$有相同的焦点$F_1$、$F_2$, $M$是椭圆$C_1$与双曲线$C_2$的公共点, 且$\\triangle MF_1F_2$的周长为$6$, 求椭圆$C_1$的方程;\\\\\n我们把具有公共焦点、公共对称轴的两段圆锥曲线弧合成的封闭曲线称为``盾圆''.\\\\\n(2) 如图, 已知``盾圆$D$''的方程为$y^2=\\begin{cases}\n4x, & 0\\le x\\le 3,\\\\ -12(x-4), & 3=latex]\n \\draw [->] (-0.5,0) -- (6,0) node [below] {$x$};\n \\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [domain = {-2*sqrt(3)}:{2*sqrt(3)}, samples = 200] plot ({\\x*\\x/4},\\x);\n \\draw [domain = {-2*sqrt(3)}:{2*sqrt(3)}, samples = 200] plot ({\\x*\\x/(-12)+4},\\x);\n \\draw [dashed] (3,-4) -- (3,4);\n \\draw (3,0) node [below left] {$3$};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0713003X" + ], "tags": [ "第七单元", "椭圆", @@ -105899,7 +106275,9 @@ "004099": { "id": "004099", "content": "已知椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 定义椭圆$C$上的点$M(x_0,y_0)$的``伴随点''为$N(\\dfrac{x_0}a,\\dfrac{y_0}b)$.\\\\\n(1) 求椭圆$C$上的点$M$的``伴随点''$N$的轨迹方程;\\\\\n(2) 如果椭圆$C$上的点$(1,\\dfrac 32)$的``伴随点''为$(\\dfrac 12,\\dfrac 3{2b})$, 对于椭圆$C$上的任意点$M$及它的``伴随点''$N$, 求$\\overrightarrow{OM}\\cdot \\overrightarrow{ON}$的取值范围;\\\\\n(3) 当$a=2$, $b=\\sqrt 3$时, 直线$l$交椭圆$C$于$A$, $B$两点, 若点$A$, $B$的``伴随点''分别是$P$, $Q$, 且以$PQ$为直径的圆经过坐标原点$O$, 求$\\triangle OAB$的面积.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -106161,7 +106539,9 @@ "004109": { "id": "004109", "content": "直线$l:(n+2)x-y+n-1=0$($n\\in \\mathbf{N}^*$)被圆$C:(x-1)^2+y^2=16$所截得的弦长为$d_n$, 则$\\displaystyle\\lim_{n\\to \\infty}d_n=$\\blank{50}.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第七单元", "圆" @@ -106459,7 +106839,9 @@ "004120": { "id": "004120", "content": "已知椭圆$\\Omega :\\dfrac{x^2}4+\\dfrac{y^2}3=1$的左右两焦点分别为$F_1,F_2$.\\\\\n(1) 若矩形$ABCD$的边$AB$在$y$轴上, 点$C,D$均在$\\Omega$上, 求该矩形绕$y$轴旋转一周所得圆柱侧面积$S$的取值范围;\\\\\n(2) 设斜率为$k$的直线$l$与$\\Omega$交于$P,Q$两点, 线段$PQ$的中点为$M(1,m)$($m>0$), 求证: $k<-\\dfrac 12$;\\\\\n(3) 过$\\Omega$上一动点$E(x_0,y_0)$作直线$l:\\dfrac{x_0x}4+\\dfrac{y_0y}3=1$, 其中$y_0\\ne 0$, 过$E$作直线$l$的垂线交$x$轴于点$R$. 问是否存在实数$\\lambda$, 使得$|EF_1|\\cdot |RF_2|=\\lambda |EF_2|\\cdot |RF_1|$恒成立? 若存在, 求出$\\lambda$的值; 若不存在, 说明理由.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -106724,7 +107106,9 @@ "004129": { "id": "004129", "content": "设椭圆$\\Gamma:\\dfrac{x^2}{a^2}+y^2=1$($a>1$)的左顶点为$A$, 过点$A$的直线$l$与$\\Gamma$相交于另一点$B$, 与$y$轴相交于点$C$. 若$|OA|=|OC|$, $|AB|=|AC|$, 则$a=$\\blank{50}.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -107342,7 +107726,9 @@ "004152": { "id": "004152", "content": "已知椭圆$x^2+\\dfrac{y^2}{b^2}=1$($0=latex]\n \\draw [->] (-2,0) -- (2.5,0) node [below] {$x$};\n \\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\filldraw (2,0) circle (0.03) node [below] {$T$};\n \\filldraw (0,1) circle (0.03) node [below left] {$B$};\n \\draw [name path = gamma] (0,0) ellipse ({sqrt(2)} and 1);\n \\path [name path = upray] (1,0) --++ ({-sqrt(2)/2},{sqrt(7)/2});\n \\path [name path = downray] (1,0) --++ ({sqrt(2)/4},{-sqrt(7)/4});\n \\path [name intersections = {of = gamma and upray, by = M}];\n \\path [name intersections = {of = gamma and downray, by = N}];\n \\draw [->] (1,0) -- (M) node [above] {$M$};\n \\draw [->] (N) node [right] {$N$} -- (1,0) node [above] {$F$};\n \\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$\\Gamma$的标准方程;\\\\\n(2) 若点$P,Q$是椭圆$\\Gamma$上异于点$B$的两点, $BP\\perp BQ$, 且满足$3\\overrightarrow {PC}=2\\overrightarrow {CQ}$的点$C$在$y$轴上, 求直线$BP$的方程;\\\\\n(3) 设$x$轴上点$T$坐标为$(2,0)$, 过椭圆$\\Gamma$的右焦点$F$作直线$l$(不与$x$轴重合)与椭圆$\\Gamma$交于$M$、$N$两点, 如图, 点$M$在$x$轴上方, 点$N$在$x$轴下方, 且$\\overrightarrow {FM}=2\\overrightarrow {NF}$, 求$|\\overrightarrow {TM}+\\overrightarrow {TN}|$的值.", - "objs": [], + "objs": [ + "K0713004X", + "K0715003X" + ], "tags": [ "第五单元", "第七单元", @@ -108153,7 +108542,9 @@ "004182": { "id": "004182", "content": "已知椭圆$\\dfrac{x^2}{6}+\\dfrac{y^2}{3}=1$上有两点$P(-2,1)$及$Q(2,-1)$, 直线$l:y=kx+b$与椭圆交于$A$、$B$两点, 与线段$PQ$交于点$C$(异于$P$、$Q$).\\\\\n(1) 当$k=1$且$\\overrightarrow{PC}=\\dfrac 12\\overrightarrow{CQ}$时, 求直线$l$的方程;\\\\\n(2) 当$k=2$时, 求四边形$PAQB$面积的取值范围.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -108695,7 +109086,10 @@ "004204": { "id": "004204", "content": "设椭圆$\\Gamma$: $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右焦点为$F(1,0)$, 短轴的一个端点$B$到$F$的距离等于焦距.\n(1) 求椭圆$\\Gamma$的标准方程;\\\\\n(2) 设$C$、$D$是四条直线$x=\\pm a$, $y=\\pm b$所围成的矩形在第一、第二象限的两个顶点, $P$是椭圆$\\Gamma$上任意一点, 若$\\overrightarrow{OP}=m\\overrightarrow{OC}+n\\overrightarrow{OD}$, 求证: $m^2+n^2$为定值;\\\\\n(3) 过点$F$的直线$l$与椭圆$\\Gamma$交于不同的两点$M$、$N$, 且满足$\\triangle BFM$与$\\triangle BFN$的面积的比值为$2$, 求直线$l$的方程.", - "objs": [], + "objs": [ + "K0713003X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -109110,7 +109504,10 @@ "004221": { "id": "004221", "content": "在圆锥$PO$中, 已知高$PO=2$, 底面圆的直径$AB=8$, $M$为母线$PB$的中点. 根据圆锥曲线的定义, 下列四个图中的截面边界曲线分别为圆(截面平行于底面)、椭圆(椭圆长轴为线段$AM$)、双曲线的一部分(双曲线所在平面垂直于$AB$)及抛物线的一部分(抛物线对称轴为$MO$所在直线), 下面四个命题:\\\\\n\\textcircled{1} 圆的面积为$4\\pi$; \\textcircled{2} 椭圆的长轴为$\\sqrt{37}$; \\textcircled{3} 双曲线两渐近线的夹角为$\\arcsin \\dfrac 35$; \\textcircled{4} 抛物线中焦点到准线的距离为$\\dfrac{8\\sqrt 5}5$中, 正确的个数为\\bracket{20}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (1,{-sqrt(3)}) node [right] {$B$} -- (0,0) node [above] {$P$} -- (-1,{-sqrt(3)}) node [left] {$A$};\n \\draw [domain = 90:270,samples = 200,dashed] plot ({1/2*sin(\\x)},{-1/6*(cos(\\x))-sqrt(3)/2});\n \\draw [domain = 90:-90,samples = 200] plot ({1/2*sin(\\x)},{-1/6*(cos(\\x))-sqrt(3)/2});\n \\draw [domain = 90:270,samples = 200,dashed] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\draw [domain = 90:-90,samples = 200] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\filldraw (0,{-sqrt(3)}) circle (0.03) node [right] {$O$} ({1/2},{-sqrt(3)/2}) node [right] {$M$};\n \\end{tikzpicture}\n \\begin{tikzpicture}\n \\draw (1,{-sqrt(3)}) node [right] {$B$} -- (0,0) node [above] {$P$} -- (-1,{-sqrt(3)}) node [left] {$A$};\n \\draw [domain = 90:270,samples = 200,dashed] plot ({2*sin(\\x)/(3+sin(\\x))},{-1/3*2*cos(\\x)/(3+sin(\\x))-2*sqrt(3)/(3+sin(\\x))});\n \\draw [domain = 90:-90,samples = 200] plot ({2*sin(\\x)/(3+sin(\\x))},{-1/3*2*cos(\\x)/(3+sin(\\x))-2*sqrt(3)/(3+sin(\\x))});\n \\draw [domain = 90:270,samples = 200,dashed] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\draw [domain = 90:-90,samples = 200] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\filldraw (0,{-sqrt(3)}) circle (0.03) node [right] {$O$} ({1/2},{-sqrt(3)/2}) node [right] {$M$};\n \\end{tikzpicture}\n \\begin{tikzpicture}\n \\draw (1,{-sqrt(3)}) node [right] {$B$} -- (0,0) node [above] {$P$} -- (-1,{-sqrt(3)}) node [left] {$A$};\n \\draw [domain = 30:90,samples = 200,dashed] plot ({1/2},{cos(\\x)/6/sin(\\x)-sqrt(3)/2/sin(\\x)});\n \\draw [domain = 150:90,samples = 200] plot ({1/2},{cos(\\x)/6/sin(\\x)-sqrt(3)/2/sin(\\x)});\n \\draw [domain = 90:270,samples = 200,dashed] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\draw [domain = 90:-90,samples = 200] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\filldraw (0,{-sqrt(3)}) circle (0.03) node [left] {$O$} ({1/2},{-sqrt(3)/2}) node [right] {$M$};\n \\end{tikzpicture}\n \\begin{tikzpicture}\n \\draw (1,{-sqrt(3)}) node [right] {$B$} -- (0,0) node [above] {$P$} -- (-1,{-sqrt(3)}) node [left] {$A$};\n \\draw [domain = 0:90,samples = 200,dashed] plot ({sin(\\x)/(1+sin(\\x))},{1/3*cos(\\x)/(1+sin(\\x))-sqrt(3)/(1+sin(\\x))});\n \\draw [domain = 180:90,samples = 200] plot ({sin(\\x)/(1+sin(\\x))},{1/3*cos(\\x)/(1+sin(\\x))-sqrt(3)/(1+sin(\\x))});\n \\draw [domain = 90:270,samples = 200,dashed] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\draw [domain = 90:-90,samples = 200] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\filldraw (0,{-sqrt(3)}) circle (0.03) node [left] {$O$} ({1/2},{-sqrt(3)/2}) node [right] {$M$};\n \\end{tikzpicture}\n\\end{center}\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", - "objs": [], + "objs": [ + "K0713001X", + "K0713002X" + ], "tags": [ "第六单元", "第七单元", @@ -109659,7 +110056,10 @@ "004242": { "id": "004242", "content": "已知点$P$为椭圆$\\dfrac{x^2}9+\\dfrac{y^2}{16}=1$上的任意一点, 点$F_1$、$F_2$分别为该椭圆的上下焦点, 设$\\alpha =\\angle PF_1F_2$, $\\beta =\\angle PF_2F_1$, 则$\\sin \\alpha +\\sin \\beta$的最大值为\\bracket{20}.\n\\fourch{$\\dfrac{3\\sqrt 7}7$}{$\\dfrac{4\\sqrt 7}7$}{$\\dfrac 89$}{$\\dfrac 32$}", - "objs": [], + "objs": [ + "K0713002X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -109946,7 +110346,9 @@ "004252": { "id": "004252", "content": "己知圆的方程为$x^2+y^2-2x-4y+4=0$, 则圆心到直线$l:3x+4y+4=0$的距离$d=$\\blank{50}.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -110867,7 +111269,11 @@ "004288": { "id": "004288", "content": "如图, 已知椭圆$M:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)经过圆$N:x^2+(y+1)^2=4$与$x$轴的两个交点和与$y$轴正半轴的交点.\\\\\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n \\draw [->] (0,-3) -- (0,2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,-1) node [below left] {$N$} circle (2) (0,0) ellipse ({sqrt(3)} and 1);\n \\draw ({sqrt(3)*cos(60)},{sin(60)}) node [above] {$P$} -- ({2*cos(-100)},{-1+2*sin(-100)}) node [below] {$Q$};\n \\filldraw (0.25,0.5) circle (0.03) node [below] {$E$};\n \\draw (-1.171,0.7368) node [left] {$A$} -- (-1.0314,0.7136) node [below] {$C$} -- (1.5314,0.2864) node [below] {$D$} -- (1.671,0.2632) node [right] {$B$};\n \\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$M$的方程;\\\\\n(2) 若点$P$为椭圆$M$上的动点, 点$Q$为圆$N$上的动点, 求线段$PQ$长的最大值;\\\\\n(3) 若不平行于坐标轴的直线$L$交椭圆$M$于$A$、$B$两点, 交圆$N$于$C$、$D$两点, 且满足$\\overrightarrow{AC}=\\overrightarrow{DB}$, 求证: 线段$AB$的中点$E$在定直线上.", - "objs": [], + "objs": [ + "K0713003X", + "K0715002X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -111269,7 +111675,9 @@ "004303": { "id": "004303", "content": "椭圆的参数方程为$\\begin{cases} x=5\\cos \\theta, \\\\ y=3\\sin \\theta \\end{cases}$($\\theta$ 为参数), 则它的两个焦点坐标是\\bracket{20}.\n\\fourch{$(\\pm 4,0)$}{$(0,\\pm 4)$}{$(\\pm 5,0)$}{$(0,\\pm 3)$}", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆", @@ -111413,7 +111821,10 @@ "004309": { "id": "004309", "content": "已知椭圆$C:\\dfrac{x^2}2+y^2=1$的左、右焦点分别为$F_1$、$F_2$.\\\\\n(1) 点$P$在椭圆$C$上运动(点$P$不在$x$轴上), 设$F_2$关于$\\angle F_1PF_2$的外角平分线所在直线的对称点为$Q$, 求$Q$的轨迹方程;\\\\\n(2) 设$M$、$N$分别是曲线$C$上的两个不同点, 且点$M$在第一象限, 点$N$在第三象限, 若$\\overrightarrow{OM}+2\\overrightarrow{ON}=2\\overrightarrow{OF_1}$, $O$为坐标原点, 求直线$MN$的斜率;\\\\\n(3) 过点$S(0,-\\dfrac 13)$的动直线$l$交曲线$C$于$A$、$B$两点, 在$y$轴上是否存在定点$T$, 使以$AB$为直径的圆恒过这个点? 若存在, 求出点$T$的坐标; 若不存在, 请说明理由.", - "objs": [], + "objs": [ + "K0713002X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -112400,7 +112811,9 @@ "004346": { "id": "004346", "content": "设三角形$ABC$是位于平面直角坐标系$xOy$的第一象限中的一个不等边三角形. 该平面上的动点$P$满足: $|PA|^2+|PB|^2+|PC|^2=|OA|^2+|OB|^2+|OC|^2$. 已知动点$P$的轨迹是一个圆, 则该圆的圆心位于三角形$ABC$的\\bracket{20}.\n\\fourch{内心}{外心}{重心}{垂心}", - "objs": [], + "objs": [ + "K0721002X" + ], "tags": [ "第七单元", "圆" @@ -112803,7 +113216,9 @@ "004361": { "id": "004361", "content": "设椭圆$\\Gamma:\\dfrac{x^2}{a^2}+{y^2}=1$($a>1$)的左顶点为$A$, 过点$A$的直线$l$与$\\Gamma$相交于另一点$B$, 与$y$轴相交于点$C$. 若$|OA|=|OC|$, $|AB|=|BC|$, 则$a=$\\blank{50}.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -116445,7 +116860,9 @@ "004487": { "id": "004487", "content": "已知直线$l:x=t$($0=latex]\n \\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [name path = ellipse] (0,0) ellipse (2 and {sqrt(2)});\n \\draw [name path = line] (0.8,-1.5) -- (0.8,1.5);\n \\draw [name intersections = {of = line and ellipse, by = {A,B}}];\n \\filldraw (A) circle (0.03) node [above right] {$A$};\n \\filldraw (B) circle (0.03) node [below right] {$B$};\n \\end{tikzpicture}\n\\end{center}\n(1) 记$F_1,F_2$是椭圆$\\Gamma$的左右焦点, 若直线$AB$过$F_2$, 当$M$到$F_1$的距离与到直线$AB$的距离相等时, 求点$M$的横坐标;\\\\\n(2) 若点$M,A$关于$y$轴对称, 当$\\triangle MAB$的面积最大时, 求直线$MB$的方程;\\\\\n(3) 设直线$MA$和$MB$与$x$轴分别交于$PQ$, 证明: $|OP|\\cdot |OQ|$为定值.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -117423,7 +117840,9 @@ "004524": { "id": "004524", "content": "以抛物线$y^2=4x$的焦点为右焦点, 且长轴为$4$的椭圆的标准方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}+\\dfrac{y^2}{15}=1$}{$\\dfrac{x^2}{16}+\\dfrac{y^2}4=1$}{$\\dfrac{x^2}4+\\dfrac{y^2}3=1$}{$\\dfrac{x^2}4+{y^2}=1$}", - "objs": [], + "objs": [ + "K0713003X" + ], "tags": [ "第七单元", "椭圆", @@ -117547,7 +117966,10 @@ "004529": { "id": "004529", "content": "已知椭圆$C_1:\\dfrac{x^2}4+{y^2}=1$, $F_1$、$F_2$为$C_1$的左、右焦点.\\\\\n(1) 求椭圆$C_1$的焦距;\\\\\n(2) 点$Q(\\sqrt 2, \\dfrac{\\sqrt 2}2)$为椭圆$C_1$的一点, 与$OQ$平行的直线$l$与椭圆$C_1$交于两点$AB$, 若$\\triangle QAB$面积为$1$, 求直线$l$的方程;\\\\\n(3) 已知椭圆$C_1$与双曲线$C_2:x^2-y^2=1$在第一象限的交点为$M(x_M,y_M)$, 椭圆 $C_1$和双曲线$C_2$上满足$|x|\\ge |x_M|$的所有点$(x,y)$组成曲线$C$. 若点$N$是曲线$C$上一动点, 求$\\overrightarrow{NF_1}\\cdot \\overrightarrow{NF_2}$的取值范围.", - "objs": [], + "objs": [ + "K0713002X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -118400,7 +118822,9 @@ "004561": { "id": "004561", "content": "已知$P$为椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}2=1$上任意一点, $Q$与$P$关于$x$轴对称, $F_1$, $F_2$为椭圆的左、右焦点, 若有$\\overrightarrow{F_1P}\\cdot \\overrightarrow{F_2P}\\le 1$, 则向量$\\overrightarrow{F_1P}$与$\\overrightarrow{F_2Q}$的夹角的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -120103,7 +120527,9 @@ "004629": { "id": "004629", "content": "已知实数$r>0$, 圆$(x-3)^2+(y-4)^2=r^2$上有且仅有两点到直线$3x-4y-2=0$的距离为$1$, 则半径$r$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -120211,7 +120637,9 @@ "004633": { "id": "004633", "content": "设$m$是正实数, 若椭圆$mx^2+(m+1)y^2=1$的两焦点的距离为$3$, 则$m$的值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{13}-3}6$}{$\\dfrac{\\sqrt{21}-3}6$}{$\\dfrac 13$}{$\\dfrac{\\sqrt{33}-3}6$}", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -120628,7 +121056,9 @@ "004648": { "id": "004648", "content": "在平面直角坐标系$xOy$中, 圆$C$的方程为$x^2+y^2-8x+15=0$, 若直线$y=kx-2$上至少存在一点, 使得以该点为圆心, $1$为半径的圆与圆$C$有公共点, 则$k$的最大值是\\blank{50}.", - "objs": [], + "objs": [ + "K0712002X" + ], "tags": [ "第七单元", "圆" @@ -122001,7 +122431,9 @@ "004701": { "id": "004701", "content": "如图, 椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点分别为$F_1$、$F_2$, 过右焦点$F_2$与$x$轴垂直的直线交椭圆于$MN$两点, 动点$P$、$Q$分别在直线$MN$与椭圆$C$上.已知$|F_1F_2|=2$, $\\triangle MNF_1$的周长为$4\\sqrt 2$.\\\\\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 1.5]\n \\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n \\draw [->] (0,-1.3) -- (0,1.3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,0) ellipse ({sqrt(2)} and 1);\n \\draw (1,{-sqrt(2)/2}) node [below] {$N$}-- (1,{sqrt(2)/2}) node [above] {$M$};\n \\filldraw (-1,0) circle (0.03) node [below] {$F_1$} (1,0) circle (0.03) node [below right] {$F_2$};\n \\draw (-1,0) -- (-1,{sqrt(2)/2}) node [above] {$Q$} -- (1,-0.4) node [right] {$P$} -- cycle;\n \\end{tikzpicture}\n\\end{center}\n(1)\t求椭圆$C$的方程;\\\\\n(2)\t若线段$PQ$的中点在$y$轴上, 求三角形$F_1QP$的面积;\\\\\n(3)\t是否存在以$F_1Q$、$F_1P$为邻边的矩形$F_1PEQ$, 使得点$E$在椭圆$C$上? 若存在, 求出所有满足条件的点$Q$的横坐标; 若不存在, 说明理由.", - "objs": [], + "objs": [ + "K0713003X" + ], "tags": [ "第七单元", "椭圆" @@ -122412,7 +122844,9 @@ "004717": { "id": "004717", "content": "椭圆$C:\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$的左、右顶点分别为$A_1,A_2$, 点$P$在$C$上($P$不与$A_1,A_2$重合)且直线$PA_2$的斜率的取值范围是$[-2,-1]$, 那么直线$PA_1$斜率的取值范围是\\bracket{20}.\n\\fourch{$[\\dfrac 12,\\dfrac 34]$}{$[\\dfrac 38,\\dfrac 34]$}{$[\\dfrac 12, 1]$}{$[\\dfrac 34,1]$}", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -122537,7 +122971,10 @@ "004722": { "id": "004722", "content": "已知椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 过定点$T(t,0)$的直线交椭圆于$P,Q$两点, 其中$t\\in (0,a)$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [name path = ell] (0,0) ellipse (2 and {sqrt(3)});\n \\draw (1,0) node [above left] {$T$} coordinate (T);\n \\path (0.4,-1.8) coordinate (U) ($(U)!2!(T)$) coordinate (V);\n \\path [name path = line] (U) -- (V);\n \\path [name intersections = {of = ell and line, by = {P,Q}}];\n \\draw ($(P)!{-0.2}!(Q)$) -- (P) node [above right] {$P$} -- (Q) node [below right ] {$Q$} -- ($(Q)!{-0.1}!(P)$) ;\n \\draw (2.5,0) node [below] {$S$} coordinate (S);\n \\draw (P) -- (S) -- (Q);\n \\end{tikzpicture}\n\\end{center}\n(1) 若椭圆短轴长为$2\\sqrt{3}$且经过点$(-1,\\dfrac 32)$, 求椭圆方程;\\\\\n(2) 对(1)中的椭圆, 若$t=\\sqrt{3}$, 求$\\triangle OPQ$面积的最大值;\\\\\n(3) 在$x$轴上是否存在点$S(s,0)$使得$\\angle PST=\\angle QST$恒成立? 如果存在, 求出$s,t$的关系; 如果不存在, 说明理由.", - "objs": [], + "objs": [ + "K0713004X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -123066,7 +123503,10 @@ "004743": { "id": "004743", "content": "如图, 中心在原点$O$的椭圆$\\Gamma$ 的右焦点为$F(2\\sqrt 3,0)$, 长轴长为$8$. 椭圆$\\Gamma$上有两点$P,Q$, 连结$OP,OQ$, 记它们的斜率为$k_{OP}$、$k_{OQ}$, 且满足$k_{OP}\\cdot k_{OQ}=-\\dfrac 14$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 0.5]\n \\draw [->] (-5,0) -- (5,0) node [below] {$x$};\n \\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [name path = directrix] ({4*sqrt(3)},-4) -- ({4*sqrt(3)},5.2);\n \\draw (0,0) ellipse (4 and 2);\n \\draw ({4*cos(acos(sqrt(7)-sqrt(3)))},{2*sin(acos(sqrt(7)-sqrt(3)))}) node [above] {$P$} coordinate (P);\n \\draw ({-4*sin(acos(sqrt(7)-sqrt(3)))},{2*cos(acos(sqrt(7)-sqrt(3)))}) node [above] {$R$} coordinate (R);\n \\draw ({4*sin(acos(sqrt(7)-sqrt(3)))},{-2*cos(acos(sqrt(7)-sqrt(3)))}) node [below] {$Q$} coordinate (Q);\n \\draw (0,0) -- (P) -- (R) -- (Q) -- (P);\n \\draw [name path = line1] (Q) -- ($(Q)!2.7!(P)$);\n \\draw [name path = line2] (R) -- ($(R)!1.7!(P)$);\n \\draw [name intersections = {of = line1 and directrix, by = N}] (N) node [right] {$N$};\n \\draw [name intersections = {of = line2 and directrix, by = M}] (M) node [right] {$M$};\n \\end{tikzpicture}\n\\end{center}\n(1)求椭圆$\\Gamma$的标准方程;\n(2)求证: ${{| OP |}^2}+{{| OQ |}^2}$为一定值, 并求出这个定值;\n(3)设直线$OQ$与椭圆$\\Gamma$的另一个交点为$R$, 直线$RP$ 和$PQ$分别与直线$x=4\\sqrt 3$ 交于点$M,N$, 若$\\triangle PQR$和$\\triangle PMN$的面积相等, 求点$P$的横坐标.", - "objs": [], + "objs": [ + "K0713003X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -218059,7 +218499,9 @@ "008854": { "id": "008854", "content": "分别根据下列条件, 求相应圆的方程.\\\\\n(1) 圆心为$C(-\\dfrac 32,3)$, 半径为$R=\\sqrt 3$;\\\\\n(2) 圆心为$C(\\sqrt 2,1)$, 过点$A(-1,\\sqrt 2)$;\\\\\n(3) 与$x$轴相交于$A(1,0)$、$B(5,0)$两点, 且半径等于$\\sqrt 5$.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -218083,7 +218525,9 @@ "008855": { "id": "008855", "content": "已知圆$(x-a)^2+(y-b)^2=r^2(r>0)$. 求在下列情况下, 实数$a,b,r$分别应满足什么条件.\n(1) 圆过原点;\\\\\n(2) 圆心在$x$轴上;\\\\\n(3) 圆与$x$轴相切;\\\\\n(4) 圆与坐标轴相切.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -218107,7 +218551,9 @@ "008856": { "id": "008856", "content": "求经过点$(5,-5)$且与圆$(x-1)^2+(y+2)^2=25$相切的直线的方程.", - "objs": [], + "objs": [ + "K0711003X" + ], "tags": [ "第七单元", "圆" @@ -218152,7 +218598,9 @@ "008858": { "id": "008858", "content": "``$A=C\\ne 0$且$B=0$''是``$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$表示圆的方程''的\\blank{50}条件.", - "objs": [], + "objs": [ + "K0710002X" + ], "tags": [ "第七单元", "圆" @@ -218176,7 +218624,9 @@ "008859": { "id": "008859", "content": "直线$Ax+By=0$与圆$x^2+y^2+Ax+By=0$的位置关系是\\blank{50}.", - "objs": [], + "objs": [ + "K0711001X" + ], "tags": [ "第七单元", "圆" @@ -218198,7 +218648,9 @@ "008860": { "id": "008860", "content": "已知$a^2x^2+(a+2)y^2+2ax+a=0$表示圆, 求实数$a$的值.", - "objs": [], + "objs": [ + "K0710002X" + ], "tags": [ "第七单元", "圆" @@ -218222,7 +218674,9 @@ "008861": { "id": "008861", "content": "已知圆过原点, 且与$x$轴、$y$轴的交点的坐标分别为$(a,0)$、$(0,b)$, 求这个圆的方程.", - "objs": [], + "objs": [ + "K0710005X" + ], "tags": [ "第七单元", "圆" @@ -218246,7 +218700,9 @@ "008862": { "id": "008862", "content": "求经过点$(5,-5)$且与圆$x^2+y^2=25$相切的直线的方程.", - "objs": [], + "objs": [ + "K0711004X" + ], "tags": [ "第七单元", "圆" @@ -218273,7 +218729,9 @@ "008863": { "id": "008863", "content": "已知动直线$kx-y+1=0$和圆$x^2+y^2=1$相交于$AB$两点, 求弦$AB$的中点的轨迹方程.", - "objs": [], + "objs": [ + "K0711005X" + ], "tags": [ "第七单元", "圆" @@ -218295,7 +218753,9 @@ "008864": { "id": "008864", "content": "已知直线$x\\cdot \\sin \\alpha +y\\cdot \\cos \\alpha +m=0(\\alpha \\in (0,\\dfrac{\\pi }2))$被圆$x^2+y^2=2$所截得的线段的长为$\\dfrac 43\\sqrt 3$, 求实数$m$的值.", - "objs": [], + "objs": [ + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -218317,7 +218777,9 @@ "008865": { "id": "008865", "content": "已知直线$l$与直线$4x-3y+18=0$垂直, 且它被圆$x^2+y^2-2x+4y-20=0$所截得的线段的长为$8$, 求直线$l$的方程.", - "objs": [], + "objs": [ + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -218339,7 +218801,9 @@ "008866": { "id": "008866", "content": "求与圆$x^2+y^2=25$外切于点$P(4,-3)$, 且半径为$1$的圆的方程.", - "objs": [], + "objs": [ + "K0710005X" + ], "tags": [ "第七单元", "圆" @@ -218390,7 +218854,9 @@ "008868": { "id": "008868", "content": "求过点$(2,-1)$, 圆心在直线$2x+y=0$上, 且与直线$x-y-1=0$相切的圆的方程.", - "objs": [], + "objs": [ + "K0709007X" + ], "tags": [ "第七单元", "圆" @@ -218415,7 +218881,9 @@ "008869": { "id": "008869", "content": "已知圆$x^2+y^2+6x-8y+25=r^2$与$x$轴相切, 求这个圆截$y$轴所得的弦长.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -218464,7 +218932,9 @@ "008871": { "id": "008871", "content": "已知定点$A(3,1)$, 动点$B$在圆$x^2+y^2=4$上, $P$在线段$AB$上, 且$BP:PA=1:2$, 求点$P$的轨迹方程.", - "objs": [], + "objs": [ + "K0721004X" + ], "tags": [ "第七单元", "圆" @@ -218486,7 +218956,9 @@ "008872": { "id": "008872", "content": "已知圆$C$与$y$轴相切, 圆心点$C$在直线$x-3y=0$上, 且直线$y=x$被圆$C$所截得的线段的长为$2\\sqrt 7$, 求圆$C$的方程.", - "objs": [], + "objs": [ + "K0709007X" + ], "tags": [ "第七单元", "圆" @@ -218508,7 +218980,10 @@ "008873": { "id": "008873", "content": "写出分别满足下列条件的椭圆的标准方程.\\\\\n(1) 焦点坐标为$(-6,0)$、$(6,0)$, 且椭圆经过点$(0,8)$;\\\\\n(2) 椭圆经过$(0,-2)$、$(\\sqrt 6,0)$两点;\\\\\n(3) 焦距等于$4$, 且椭圆经过点$P(\\dfrac{2\\sqrt 6}3,-\\dfrac{2\\sqrt 6}3)$.", - "objs": [], + "objs": [ + "K0713003X", + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -218551,7 +219026,9 @@ "008875": { "id": "008875", "content": "若方程$\\dfrac{x^2}m+\\dfrac{y^2}{{m^2}-2}=1$表示椭圆, 求实数$m$的取值范围.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -218573,7 +219050,9 @@ "008876": { "id": "008876", "content": "若椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$的两个焦点分别为$F_1F_2$, 点$P$为此椭圆上的任意一点, 求$\\triangle PF_1F_2$的周长.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -218595,7 +219074,9 @@ "008877": { "id": "008877", "content": "已知椭圆的方程为$\\dfrac{x^2}{16}+\\dfrac{y^2}{m^2}=1(m>0)$.如果此椭圆的焦点在$x$轴上, 那么它的焦距为\\bracket{20}.\n\\fourch{$2\\sqrt {16-m^2}$}{$2\\sqrt {4-m}$}{$2\\sqrt {m^2-8}$}{$2\\sqrt {m-4}$}", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -218617,7 +219098,9 @@ "008878": { "id": "008878", "content": "若方程$16x^2+ky^2=16k$表示焦点在$y$轴上的椭圆, 则实数$k$满足\\bracket{20}.\n\\fourch{$k>16$}{$k=16$}{$k<16$}{$0b>0)$上, 则\\bracket{20}.\n\\twoch{点$(4,-3)$不在椭圆上}{点$(3,4)$在椭圆上}{点$(-4,-3)$不在椭圆上}{点$(-4,3)$在椭圆上}", - "objs": [], + "objs": [ + "K0714003X" + ], "tags": [ "第七单元", "椭圆" @@ -218751,7 +219245,10 @@ "008884": { "id": "008884", "content": "过点$(3,-2)$且与椭圆$4x^2+9y^2=36$有相同焦点的椭圆的标准方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{15}+\\dfrac{y^2}{10}=1$}{$\\dfrac{x^2}{{{15}^2}}+\\dfrac{y^2}{{{10}^2}}=1$}{$\\dfrac{x^2}{10}+\\dfrac{y^2}{15}=1$}{$\\dfrac{x^2}{{{10}^2}}+\\dfrac{y^2}{{{15}^2}}=1$}", - "objs": [], + "objs": [ + "K0713002X", + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -218773,7 +219270,9 @@ "008885": { "id": "008885", "content": "若椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$的弦$AB$被点$P(1,1)$平分, 则$AB$所在直线的方程为 \\bracket{20}.\n\\fourch{$9x+4y-13=0$}{$4x+9y-13=0$}{$x+2y-3=0$}{$x+3y-3=0$}", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -218795,7 +219294,9 @@ "008886": { "id": "008886", "content": "画出长轴长和短轴长分别为$2$厘米、$1.5$厘米的椭圆的草图.若要把一个边长分别为$2$米和$1.5$米的矩形木板锯成椭圆形, 使它的长轴长和短轴长分别为$2$米、$1.5$米, 请用简便的方法在木板上画出这个椭圆的草图.", - "objs": [], + "objs": [ + "K0713001X" + ], "tags": [ "第七单元", "椭圆" @@ -218817,7 +219318,10 @@ "008887": { "id": "008887", "content": "已知椭圆以原点为中心, 长轴长是短轴长的$2$倍, 且过点$(-2,-4)$, 求此椭圆的标准方程.", - "objs": [], + "objs": [ + "K0713003X", + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -218860,7 +219364,9 @@ "008889": { "id": "008889", "content": "已知椭圆的一个顶点和一个焦点分别是直线$x+3y-6=0$与两坐标轴的交点, 求此椭圆的标准方程.", - "objs": [], + "objs": [ + "K0713003X" + ], "tags": [ "第七单元", "椭圆" @@ -218882,7 +219388,10 @@ "008890": { "id": "008890", "content": "已知椭圆$C$的焦点分别为$F_1(-2\\sqrt 2,0)$、$F_2(2\\sqrt 2,0)$, 长轴长为$6$, 直线$y=x+2$交椭圆$C$于$AB$两点, 求线段$AB$的中点的坐标.", - "objs": [], + "objs": [ + "K0713003X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -218904,7 +219413,9 @@ "008891": { "id": "008891", "content": "已知点$P$是椭圆$\\dfrac{x^2}{100}+\\dfrac{y^2}{36}=1$上一点, 它到椭圆的左焦点$F_1$的距离是它到右焦点$F_2$的距离的$3$倍.\\\\\n(1) 分别求点$P$与点$F_1$、点$P$与点$F_2$的距离;\\\\\n(2) 求点$P$的坐标.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -218926,7 +219437,10 @@ "008892": { "id": "008892", "content": "以椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$的两个焦点及短轴的两个端点为四个顶点的椭圆方程为\\blank{50}.", - "objs": [], + "objs": [ + "K0714003X", + "K0713003X" + ], "tags": [ "第七单元", "椭圆" @@ -218948,7 +219462,9 @@ "008893": { "id": "008893", "content": "如果点$P$是椭圆$\\dfrac{x^2}{36}+\\dfrac{y^2}{20}=1$上一个动点, $F_1$是椭圆的左焦点, 那么$|PF_1|$的最大值是\\blank{50}, $|PF_1|$的最小值是\\blank{50}.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -218970,7 +219486,9 @@ "008894": { "id": "008894", "content": "如果直线$y=kx+1$与椭圆$\\dfrac{x^2}5+\\dfrac{y^2}m=1$恒有公共点, 那么实数$m$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0713001X" + ], "tags": [ "第七单元", "椭圆" @@ -218992,7 +219510,9 @@ "008895": { "id": "008895", "content": "椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$与$\\dfrac{x^2}{9-k}+\\dfrac{y^2}{25-k}=1(0b>0)$与直线$x+2y-2=0$交于$AB$两点, $|AB|=\\sqrt 5$, 且$AB$的中点的坐标为$(m,\\dfrac 12)$, 求此椭圆的方程.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -219462,7 +219987,10 @@ "008916": { "id": "008916", "content": "求以椭圆$\\dfrac{x^2}8+\\dfrac{y^2}5=1$的焦点为顶点, 以椭圆的顶点为焦点的双曲线的方程.", - "objs": [], + "objs": [ + "K0713002X", + "K0714003X" + ], "tags": [ "第七单元", "椭圆", @@ -219981,7 +220509,9 @@ "008939": { "id": "008939", "content": "直线$x-\\sqrt 3y=0$绕原点按逆时针方向旋转$30^{\\circ }$后所得的直线与圆$(x-2)^2+y^2=3$的位置关系是\\bracket{20}.\n\\twoch{直线过圆心}{直线与圆相交, 但不过圆心}{直线与圆相切}{直线与圆无公共点}", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -220005,7 +220535,9 @@ "008940": { "id": "008940", "content": "分别求下列各圆的标准方程.\\\\\n(1) 圆心在直线$y=-x$上, 且过$(2,0)$、$(0,-4)$两点;\\\\\n(2) 圆心在直线$2x+y=0$上, 且与直线$x+y-1=0$相切于点$(2,-1)$.", - "objs": [], + "objs": [ + "K0709007X" + ], "tags": [ "第七单元", "圆" @@ -220027,7 +220559,10 @@ "008941": { "id": "008941", "content": "已知圆$O$的方程是$x^2+y^2=1$, 直线$l$与圆$O$相切.\\\\\n(1) 若直线$l$的斜率等于$1$, 求直线$l$的方程;\\\\\n(2) 若直线$l$在$y$轴上的截距为$\\sqrt 2$, 求直线$l$的方程.", - "objs": [], + "objs": [ + "K0711001X", + "K0711004X" + ], "tags": [ "第七单元", "圆" @@ -220051,7 +220586,9 @@ "008942": { "id": "008942", "content": "已知圆$x^2+y^2+x-6y+m=0$与直线$x+2y-3=0$相交于$PQ$两点, $O$为坐标原点, 若$OP\\perp OQ$, 求实数$m$的值.", - "objs": [], + "objs": [ + "K0710006X" + ], "tags": [ "第七单元", "圆" @@ -220075,7 +220612,9 @@ "008943": { "id": "008943", "content": "已知圆$C$的圆心在直线$l_1$: $x-3y=0$上, 圆$C$与$y$轴相切, 且直线$l_2$: $x-y=0$被圆$C$所截得的线段的长为$2\\sqrt 7$, 求圆$C$的方程.", - "objs": [], + "objs": [ + "K0710005X" + ], "tags": [ "第七单元", "圆" @@ -220120,7 +220659,9 @@ "008945": { "id": "008945", "content": "已知$F_1,F_2$为椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}9=1$的两个焦点, 过点$F_2$的直线交椭圆于$AB$两点, 且$|AB|=5$, 求$|AF_1|+|BF_1|$的值.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -220142,7 +220683,9 @@ "008946": { "id": "008946", "content": "已知倾斜角为$\\dfrac{\\pi }4$的直线交椭圆$\\dfrac{x^2}4+y^2=1$于$A,B$两点, 求线段$AB$的中点$P$的轨迹方程.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -220164,7 +220707,9 @@ "008947": { "id": "008947", "content": "已知过点$M(-2,0)$的直线$l$与椭圆$x^2+2y^2=2$交于$P_1,P_2$两点, 线段$P_1P_2$的中点为$P$, 设直线$l$的斜率为$k_1(k_1\\ne 0)$, 直线$OP$的斜率为$k_2$, 求证: $k_1\\cdot k_2$的值为定值.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -220186,7 +220731,9 @@ "008948": { "id": "008948", "content": "已知椭圆$\\dfrac{x^2}{45}+\\dfrac{y^2}{20}=1$的焦点分别是$F_1,F_2$, 过中心$O$作直线与椭圆相交于$A,B$两点, 若要使$\\triangle ABF_2$的面积是$20$, 求直线$AB$的方程.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -220208,7 +220755,9 @@ "008949": { "id": "008949", "content": "椭圆$x^2+4y^2=4$的长轴上的一个顶点为$A$, 以$A$为直角顶点作一个内接于此椭圆的等腰直角三角形, 求这个三角形的面积.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -220407,7 +220956,9 @@ "008958": { "id": "008958", "content": "设圆的方程是$(x-a)^2+(y+b)^2=a^2+b^2$, 其中$a>0,b>0$, 给出下列三种说法:\n\\textcircled{1} 该圆的圆心$(a,b)$;\n\\textcircled{2} 该圆过原点;\n\\textcircled{3} 该圆与$x$轴相交于两个不同点.\n其中\\bracket{20}.\n\\fourch{只有\\textcircled{1}与\\textcircled{2}正确}{只有\\textcircled{1}与\\textcircled{3}正确}{只有\\textcircled{2}与\\textcircled{3}正确}{\\textcircled{1}、\\textcircled{2}与\\textcircled{3}都正确}", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -220429,7 +220980,9 @@ "008959": { "id": "008959", "content": "若椭圆$\\dfrac{x^2}4+\\dfrac{y^2}{a^2}=1$与双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}2=1$有相同的焦点, 则实数$a$为\\bracket{20}.\n\\fourch{1}{$-1$}{$\\pm 1$}{不确定}", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆", @@ -220498,7 +221051,9 @@ "008962": { "id": "008962", "content": "命题: 椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$与双曲线$\\dfrac{x^2}{11}-\\dfrac{y^2}5=1$的焦距相等.试将此命题推广到一般情形, 使已知命题成为推广后命题的一个特例:\\blank{50}.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆", @@ -220568,7 +221123,9 @@ "008965": { "id": "008965", "content": "设$F_1,F_2$为椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$的两个焦点, $P$为椭圆上任意一点, 已知$P,F_1,F_2$是一个直角三角形的三个顶点, 且$|PF_1|>|PF_2|$, 求$\\dfrac{|PF_1|}{|PF_2|}$的值.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -222963,7 +223520,9 @@ "009076": { "id": "009076", "content": "圆的方程为$x^2+y^2+kx+2y+k^2=0$, 当圆的面积最大时, 圆心坐标是\\blank{50}.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -223049,7 +223608,9 @@ "009080": { "id": "009080", "content": "已知$F_1,F_2$是定点, $|F_1F_2|=6$.若动点$M$满足$|MF_1|+|MF_2|=6$, 则动点$M$的轨迹是\\bracket{20}.\n\\fourch{直线}{线段}{圆}{椭圆}", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -223209,7 +223770,9 @@ "009087": { "id": "009087", "content": "已知椭圆$\\dfrac 12x^2+y^2=1$和椭圆外一点$(0,2)$, 过这点引直线与椭圆交于$A,B$两点, 求弦$AB$的中点$P$的轨迹方程.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -223253,7 +223816,9 @@ "009089": { "id": "009089", "content": "直线$x-y+1=0$与椭圆$mx^2+ny^2=1(m,n>0)$相交于$AB$两点, 弦$AB$的中点的横坐标是$-\\dfrac 13$.求双曲线$\\dfrac{y^2}{m^2}-\\dfrac{x^2}{n^2}=1$的两条渐近线所夹锐角的大小.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆", @@ -223385,7 +223950,9 @@ "009095": { "id": "009095", "content": "点$A$是圆$C$: $x^2+y^2+ax+4y-5=0$上一点.若点$A$关于直线$x+2y-1=0$的对称点也在圆$C$上, 则实数$a=$\\blank{50}.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -223407,7 +223974,9 @@ "009096": { "id": "009096", "content": "若点$P$在圆$x^2+y^2+4x-6y+12=0$上, 点$Q$在直线$4x+3y-21=0$上, 则$|PQ|$的最小值为\\blank{50}.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -223606,7 +224175,9 @@ "009105": { "id": "009105", "content": "已知抛物线$y^2=4x$与椭圆$\\dfrac{x^2}9+\\dfrac{y^2}k=1$有公共焦点$F_1$, 椭圆的另一焦点为$F_2$, $P$是这两条曲线的一个交点, 求$\\triangle PF_1F_2$的周长.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆", @@ -223693,7 +224264,10 @@ "009109": { "id": "009109", "content": "已知直线$l$过点$A(-3,1)$, 且倾斜角为$45^{\\circ }$.直线$l$与焦点为$(-\\sqrt 6,0)$、$(\\sqrt 6,0)$的椭圆交于$B,C$两点, 且点$A$为线段$BC$的中点.是否存在满足上述条件的椭圆? 若存在, 求椭圆的方程; 若不存在, 请说明理由.", - "objs": [], + "objs": [ + "K0713004X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -239425,7 +239999,9 @@ "009806": { "id": "009806", "content": "求以$C(3,4)$为圆心, 且过点$M(1,-3)$的圆的方程.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -239447,7 +240023,9 @@ "009807": { "id": "009807", "content": "求以$C(-1,2)$为圆心, 且与直线$2x-3y-5=0$相切的圆的方程.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -239469,7 +240047,9 @@ "009808": { "id": "009808", "content": "一个圆与$y$轴相切于点$(0,4)$, 且在$x$轴正半轴上截得长为$6$的弦. 求此圆的方程.", - "objs": [], + "objs": [ + "K0709007X" + ], "tags": [ "第七单元", "圆" @@ -239491,7 +240071,9 @@ "009809": { "id": "009809", "content": "求经过$A(3,2)$、$B(1,1)$、$C(2,-1)$三点的圆的方程.", - "objs": [], + "objs": [ + "K0710004X" + ], "tags": [ "第七单元", "圆" @@ -239555,7 +240137,10 @@ "009812": { "id": "009812", "content": "(1) 求经过点$(-3,4)$且与圆$x^2+y^2=25$相切的直线的方程;\\\\\n(2) 求经过点$(2,4)$且与圆$x^2+y^2=4$相切的直线的方程.", - "objs": [], + "objs": [ + "K0711003X", + "K0711004X" + ], "tags": [ "第七单元", "圆" @@ -239577,7 +240162,9 @@ "009813": { "id": "009813", "content": "当$a$为何值时, 直线$x+y-a=0$与圆$x^2+y^2=2$分别有如下位置关系:\\\\\n(1) 相交;\\\\\n(2) 相切;\\\\ \n(3) 相离.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -239599,7 +240186,9 @@ "009814": { "id": "009814", "content": "已知直线$l$经过点$P(6,-4)$且被圆$x^2+y^2=20$截得长为$6\\sqrt 2$的弦, 求$l$的方程.", - "objs": [], + "objs": [ + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -239621,7 +240210,9 @@ "009815": { "id": "009815", "content": "已知圆$C_1:(x-2)^2+(y-2)^2=1$和圆$C2:x^2+(y-m)^2=m^2$($m>0$), 当$m$为何值时, 圆$C_1$与圆$C2$分别内切、相交?", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -239643,7 +240234,9 @@ "009816": { "id": "009816", "content": "求与圆$x^2+y^2=25$外切于点$P(4,-3)$且半径为$1$的圆的方程.", - "objs": [], + "objs": [ + "K0710005X" + ], "tags": [ "第七单元", "圆" @@ -239670,7 +240263,10 @@ "009817": { "id": "009817", "content": "已知圆$x^2+y^2-2x+2y-3=0$和圆$x^2+y^2+4x-1=0$相交于$A$、$B$两点, 求公共弦$AB$的长.", - "objs": [], + "objs": [ + "K0712004X", + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -239713,7 +240309,9 @@ "009819": { "id": "009819", "content": "分别写出满足下列条件的椭圆的标准方程:\\\\\n(1) 焦点在$y$轴上, 焦距为$2\\sqrt {15}$, 且经过点$(0,-4)$;\\\\\n(2) 焦距为$4$, 且经过点$(\\sqrt 5,0)$.", - "objs": [], + "objs": [ + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -239735,7 +240333,9 @@ "009820": { "id": "009820", "content": "已知下列椭圆的方程, 分别求椭圆的长轴长、短轴长、焦点坐标和顶点坐标:\\\\\n(1) $\\dfrac{x^2}4+\\dfrac{y^2}3=1$;\\\\\n(2) $25x^2+4y^2=100$.", - "objs": [], + "objs": [ + "K0714003X" + ], "tags": [ "第七单元", "椭圆" @@ -239757,7 +240357,9 @@ "009821": { "id": "009821", "content": "用离心率作为指标衡量, 下列每组两个椭圆中哪一个更接近圆?\\\\\n(1) $\\dfrac{x^2}4+\\dfrac{y^2}9=1$与$\\dfrac{x^2}{16}+\\dfrac{y^2}{12}=1$;\\\\\n(2) $x^2+9y^2=36$与$5x^2+3y^2=30$.", - "objs": [], + "objs": [ + "K0714005X" + ], "tags": [ "第七单元", "椭圆" @@ -239779,7 +240381,9 @@ "009822": { "id": "009822", "content": "若一椭圆以原点为中心, 一个焦点的坐标为$(\\sqrt 2,0)$, 且长轴长是短轴长的$\\sqrt 3$倍. 求该椭圆的标准方程.", - "objs": [], + "objs": [ + "K0713003X" + ], "tags": [ "第七单元", "椭圆" @@ -239801,7 +240405,9 @@ "009823": { "id": "009823", "content": "已知$P$是椭圆$\\dfrac{x^2}{36}+\\dfrac{y^2}{20}=1$上一个动点, $F_1$是椭圆的左焦点. 求$|PF_1|$的最大值和最小值.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -239823,7 +240429,9 @@ "009824": { "id": "009824", "content": "点$P$在焦点为$F_1$、$F_2$的椭圆$\\dfrac{x^2}{45}+\\dfrac{y^2}{20}=1$上, 且$\\angle F_1PF_2=90^\\circ$. 求$|PF1|\\cdot |PF_2|$的值.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -239845,7 +240453,9 @@ "009825": { "id": "009825", "content": "已知直线$l: y=mx-2$与椭圆$C: \\dfrac{x^2}4+\\dfrac{y^2}3=1$相交于两个不同的点, 求实数$m$的取值范围.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -240284,7 +240894,9 @@ "009845": { "id": "009845", "content": "以原点为圆心、$1$为半径作一个圆. 设定点$A$的坐标为$(2,0)$, $B$为圆上任意一点, $M$为线段$AB$的中点. 求点$M$轨迹的参数方程.", - "objs": [], + "objs": [ + "K0721004X" + ], "tags": [ "第七单元", "圆", @@ -243935,7 +244547,10 @@ "010003": { "id": "010003", "content": "已知椭圆方程$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点为$F_1(-\\sqrt{2},0)$、$F_2(\\sqrt{2},0)$, $A$为椭圆的下顶点, $M$为直线$l:x+y-4\\sqrt{2}=0$上一点.\\\\\n(1) 若$a=2$, $AM$的中点在$x$轴上, 求点$M$的坐标;\\\\\n(2) 直线$l$交$y$轴于点$B$, 直线$AM$经过$F_2$, 若$\\triangle ABM$有一个内角的余弦值为$\\dfrac 35$, 求$b$的值;\\\\\n(3) 若$\\Gamma$上存在点$P$到直线$l$的距离为$d$, 且满足$d+|PF_1|+|PF_2|=6$, 当$a$变化时, 求$d$的最小值.", - "objs": [], + "objs": [ + "K0713003X", + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -258942,7 +259557,9 @@ "010646": { "id": "010646", "content": "根据下列条件, 分别求圆的方程:\\\\\n(1) 圆心为$C(-\\dfrac 32, 3)$, 半径$r=\\sqrt 3$;\\\\\n(2) 圆心为$C(\\sqrt 2, 1)$, 过点$A(-1, \\sqrt 2)$;\\\\\n(3) 与$x$轴相交于$A(1, 0)$、$B(5, 0)$两点, 且半径等于$\\sqrt 5$.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -258966,7 +259583,9 @@ "010647": { "id": "010647", "content": "已知圆$(x-a)^2+(y-b)^2=r^2$($r>0$), 求在下列情况下, 实数$a$、$b$、$r$应分别满足什么条件:\\\\\n(1) 圆过原点;\\\\\n(2) 圆心在$x$轴上;\\\\\n(3) 圆与$x$轴相切;\\\\\n(4) 圆与两坐标轴均相切.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -258990,7 +259609,9 @@ "010648": { "id": "010648", "content": "求过点$M(5, 2)$、$N(3, 2)$, 且圆心在直线$y=2x-3$上的圆的方程.", - "objs": [], + "objs": [ + "K0711003X" + ], "tags": [ "第七单元", "圆" @@ -259014,7 +259635,9 @@ "010649": { "id": "010649", "content": "已知$a^2x^2+(a+2)y^2+2ax+a=0$表示圆, 求实数$a$的值.", - "objs": [], + "objs": [ + "K0710002X" + ], "tags": [ "第七单元", "圆" @@ -259038,7 +259661,9 @@ "010650": { "id": "010650", "content": "直线$l$与圆$x^2+y^2+2x-4y+a=0$($a<3$)相交于$A$、$B$两点, 且弦$AB$的中点为$(0, 1)$. 求直线$l$的方程.", - "objs": [], + "objs": [ + "K0710006X" + ], "tags": [ "第七单元", "圆" @@ -259060,7 +259685,9 @@ "010651": { "id": "010651", "content": "已知圆过原点, 且与$x$轴、$y$轴的交点的坐标分别为$(a, 0)$、$(0, b)$, 其中$ab\\ne 0$.求这个圆的方程.", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -259084,7 +259711,9 @@ "010652": { "id": "010652", "content": "判断直线$x\\cos \\theta +y\\sin \\theta =r$与圆$x^2+y^2=r^2$的位置关系.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -259106,7 +259735,9 @@ "010653": { "id": "010653", "content": "已知直线$2x+3y+1=0$和圆$x^2+y^2-2x-3=0$相交于$A$、$B$两点, 求弦$AB$的垂直平分线的方程.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -259128,7 +259759,9 @@ "010654": { "id": "010654", "content": "求与圆$x^2+y^2=25$内切于点$P(3, -4)$且半径为$1$的圆的方程.", - "objs": [], + "objs": [ + "K0710005X" + ], "tags": [ "第七单元", "圆" @@ -259154,7 +259787,9 @@ "010655": { "id": "010655", "content": "已知圆$C$过点$(-4, 0)$且与圆$x^2+y^2-4x-6y=0$相切于原点, 求圆$C$的方程.", - "objs": [], + "objs": [ + "K0710005X" + ], "tags": [ "第七单元", "圆" @@ -259176,7 +259811,9 @@ "010656": { "id": "010656", "content": "圆拱桥的一个圆拱如图所示, 该圆拱的跨度$AB$为$20\\text{m}$, 拱高$OP$为$4\\text{m}$, 在建造过程中每隔$4\\text{m}$需用一个支柱支撑. 求支柱$A_2B_2$的高度. (结果精确到$0.01\\text{m}$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [domain = -10:10, samples= 200, very thick] plot (\\x,{sqrt(210.25-pow(\\x,2))-10.5});\n\\draw (-10,0) node [left] {$A$} -- (10,0) node [right] {$B$};\n\\draw (0,0) node [below] {$O$} -- (0,4) node [above] {$P$};\n\\draw [very thick](-6,0) node [below] {$A_1$} -- (-6,{sqrt(210.25-pow(6,2))-10.5}) node [above] {$B_1$};\n\\draw [very thick](-2,0) node [below] {$A_2$} -- (-2,{sqrt(210.25-pow(2,2))-10.5}) node [above] {$B_2$};\n\\draw [very thick](2,0) node [below] {$A_3$} -- (2,{sqrt(210.25-pow(2,2))-10.5}) node [above] {$B_3$};\n\\draw [very thick](6,0) node [below] {$A_4$} -- (6,{sqrt(210.25-pow(6,2))-10.5}) node [above] {$B_4$};\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0709003X" + ], "tags": [ "第七单元", "圆" @@ -259198,7 +259835,9 @@ "010657": { "id": "010657", "content": "给定点$A(2, 3)$与圆$C: x^2+y^2=25$, 求圆$C$的过点$A$最短弦所在直线的方程.", - "objs": [], + "objs": [ + "K0709006X" + ], "tags": [ "第七单元", "圆" @@ -259220,7 +259859,9 @@ "010658": { "id": "010658", "content": "一个圆过点$(2, -1)$, 圆心在直线$2x+y=0$上, 且与直线$x-y-1=0$相切. 求这个圆的方程.", - "objs": [], + "objs": [ + "K0709007X" + ], "tags": [ "第七单元", "圆" @@ -259244,7 +259885,9 @@ "010659": { "id": "010659", "content": "已知圆$x^2+y^2+6x-8y+25=r^2$与$x$轴相切, 求这个圆截$y$轴所得的弦长.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -259268,7 +259911,9 @@ "010660": { "id": "010660", "content": "求圆$C: x^2+y^2+4x+2y-3=0$关于点$M(1, 1)$对称的圆的方程.", - "objs": [], + "objs": [ + "K0710003X" + ], "tags": [ "第七单元", "圆" @@ -259290,7 +259935,9 @@ "010661": { "id": "010661", "content": "已知动直线$kx-y+1=0$(其中$k\\in \\mathbf{R}$)和圆$x^2+y^2=4$相交于$A$、$B$两点, 求弦$AB$的中点的轨迹方程.", - "objs": [], + "objs": [ + "K0711005X" + ], "tags": [ "第七单元", "圆" @@ -259312,7 +259959,9 @@ "010662": { "id": "010662", "content": "求经过点$(5, -5)$且与圆$x^2+y^2=25$相切的直线的方程.", - "objs": [], + "objs": [ + "K0711004X" + ], "tags": [ "第七单元", "圆" @@ -259362,7 +260011,9 @@ "010664": { "id": "010664", "content": "已知直线$l: x-y+4=0$与圆$C: (x-1)^2+(y-1)^2=2$, 求圆$C$上各点到直线$l$的距离的最大值.", - "objs": [], + "objs": [ + "K0711002X" + ], "tags": [ "第七单元", "圆" @@ -259405,7 +260056,9 @@ "010666": { "id": "010666", "content": "$400\\text{m}$标准跑道的内圈如图所示($400\\text{m}$标准跑道最内圈的长度为$400\\text{m}$), 其中左右两端均是半径为$36\\text{m}$的半圆弧.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\filldraw [gray!50] (0,0) -- (8.7,0) arc (-90:90:3.6) --++ (-8.7,0) arc (90:270:3.6) -- cycle; \n\\draw (0,0) -- (8.7,0) arc (-90:90:3.6) --++ (-8.7,0) arc (90:270:3.6); \n\\end{tikzpicture}\n\\end{center}\n(1) 求每条直道的长度; ($\\pi$取$3.14$, 结果精确到$1\\text{m}$)\\\\\n(2) 建立适当的平面直角坐标系, 写出上半部分跑道所对应的函数表达式.", - "objs": [], + "objs": [ + "K0721002X" + ], "tags": [ "第七单元", "圆" @@ -259427,7 +260080,9 @@ "010667": { "id": "010667", "content": "若方程$16x^2+ky^2=16k$表示焦点在$y$轴上的椭圆, 求实数$k$的取值范围.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -259449,7 +260104,10 @@ "010668": { "id": "010668", "content": "设$F$是椭圆的一个焦点, $B_1B_2$是椭圆的短轴, $\\angle B_1FB_2=60^\\circ$. 求椭圆的离心率.", - "objs": [], + "objs": [ + "K0714003X", + "K0714005X" + ], "tags": [ "第七单元", "椭圆" @@ -259471,7 +260129,9 @@ "010669": { "id": "010669", "content": "已知椭圆的一个焦点是$F_1(-3, 0)$, 且经过点$P(2, \\sqrt 2)$. 求这个椭圆的标准方程.", - "objs": [], + "objs": [ + "K0713004X" + ], "tags": [ "第七单元", "椭圆" @@ -259493,7 +260153,9 @@ "010670": { "id": "010670", "content": "直线$y=2x+b$被椭圆$4x^2+y^2=16$所截得的弦长为$\\sqrt{35}$, 求实数$b$的值.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -259515,7 +260177,9 @@ "010671": { "id": "010671", "content": "若对于任意实数$k$, 直线$y=kx+1$与椭圆$\\dfrac{x^2}5+\\dfrac{y^2}m=1$恒有公共点. 求实数$m$的取值范围.", - "objs": [], + "objs": [ + "K0713001X" + ], "tags": [ "第七单元", "椭圆" @@ -259537,7 +260201,9 @@ "010672": { "id": "010672", "content": "已知$P$是椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$上的点, $F_1$、$F_2$是椭圆的两个焦点.\\\\\n(1) 若$\\angle F_1PF_2=60^\\circ$, 求$\\triangle PF_1F_2$的面积;\\\\\n(2) 若$\\triangle PF_1F_2$的面积为$9$, 求$\\angle F_1PF_2$的大小.", - "objs": [], + "objs": [ + "K0715002X" + ], "tags": [ "第七单元", "椭圆" @@ -259559,7 +260225,10 @@ "010673": { "id": "010673", "content": "水星的运行轨道是以太阳的中心为一个焦点的椭圆, 轨道上离太阳中心最近的距离约为$4.7\\times 10^8\\text{km}$, 最远的距离约为$7.05\\times 10^8\\text{km}$. 以这个轨道的中心为原点, 以太阳中心及轨道中心所在直线为$x$轴, 建立平面直角坐标系. 求水星运行轨道的方程. (长半轴的长和短半轴的长精确到$0. 1\\times 10^8\\text{km}$)", - "objs": [], + "objs": [ + "K0713003X", + "K0715001X" + ], "tags": [ "第七单元", "椭圆" @@ -259941,7 +260610,9 @@ "010690": { "id": "010690", "content": "写出椭圆方程推导过程中的``反过来推演'', 即验证:若点$M$以方程$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的解$(x, y)$为坐标, 则点$M$一定在以$F_1(-c, 0)$与$F_2(c, 0)$为焦点的椭圆上, 这里$c=\\sqrt{a^2-b^2}$.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆" @@ -260213,7 +260884,9 @@ "010702": { "id": "010702", "content": "点$P$在椭圆$\\dfrac{x^2}4+y^2=1$上运动, 求它到直线$l: x+2y-2=0$的距离的最大值.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆" @@ -260235,7 +260908,9 @@ "010703": { "id": "010703", "content": "点$P$到定点$F(2, 0)$的距离与它到直线$x=8$的距离之比为$k$, 请分别给出$k$的某个值, 使得轨迹是椭圆、双曲线和抛物线.", - "objs": [], + "objs": [ + "K0713002X" + ], "tags": [ "第七单元", "椭圆", @@ -260259,7 +260934,9 @@ "010704": { "id": "010704", "content": "已知椭圆$C: \\dfrac{x^2}4+\\dfrac{y^2}3=1$, 试确定$m$的取值范围, 使该椭圆上有两个不同的点关于直线$l: y=4x+m$对称.", - "objs": [], + "objs": [ + "K0715003X" + ], "tags": [ "第七单元", "椭圆"