From d9b18548c993ae2f273704436b741ff2ddd2cef8 Mon Sep 17 00:00:00 2001
From: wangweiye7840 <kj_wangweiye@126.com>
Date: Fri, 7 Jul 2023 13:25:47 +0800
Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A5=E7=A9=BA=E4=B8=AD=E8=AF=BE?=
 =?UTF-8?q?=E5=A0=82=E9=80=89=E5=BF=85=E4=B8=80=E7=9B=B4=E7=BA=BF=E5=8D=95?=
 =?UTF-8?q?=E5=85=83=E9=A2=98=E7=9B=AE=E5=B9=B6=E5=AF=B9=E5=BA=94=E5=8D=95?=
 =?UTF-8?q?=E5=85=83?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 题库0.3/Problems.json | 1122 +++++++++++++++++++++++++++++++++++++++++
 1 file changed, 1122 insertions(+)

diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index 2712abcd..292bf08d 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -483959,6 +483959,1128 @@
         "space": "4em",
         "unrelated": []
     },
+    "018831": {
+        "id": "018831",
+        "content": "已知直线$l$经过点$A(-1,-1)$、$B(2,3)$, 求它的斜率与倾斜角.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018832": {
+        "id": "018832",
+        "content": "已知直线$l$经过点$A(-1,-1)$、$B(2,-3)$, 求它的倾斜角.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018833": {
+        "id": "018833",
+        "content": "求一次函数$y=k x+b$($k \\neq 0$)所表示直线的斜率.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018834": {
+        "id": "018834",
+        "content": "已知三点$A(-1,2)$、$B(3,4)$、$C(x, y)$共线, 求点$C$的坐标$x$与$y$所满足的一个关系式.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018835": {
+        "id": "018835",
+        "content": "已知$a \\in \\mathbf{R}$, 若过点$A(2 a, 3)$、$B(2,-1)$的直线的倾斜角是钝角, 求$a$的取值范围.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018836": {
+        "id": "018836",
+        "content": "已知直线$l$过点$A(0,2)$、$B(-\\sqrt{3}, 3 m^2+12 m+13)$($m \\in \\mathbf{R}$), 求直线$l$的倾斜角的取值范围.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018837": {
+        "id": "018837",
+        "content": "求经过点$P(-1,1)$、$Q(2,3)$的直线$l$的点斜式方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018838": {
+        "id": "018838",
+        "content": "求倾斜角是$\\dfrac{2 \\pi}{3}$且在$x$轴上的截距是 $-3$ 的直线$l$的点斜式方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018839": {
+        "id": "018839",
+        "content": "已知直线$l$在$x$轴、$y$轴上的截距分别是$-2$、$4$, 求直线$l$的方程, 并判断点$A(2,6)$是否在直线$l$上.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018840": {
+        "id": "018840",
+        "content": "已知直线$l$过点$P(-1,2)$.\\\\\n(1) 若直线$l$在两坐标轴上的截距和为零, 求直线$l$的方程;\\\\\n(2) 设直线$l$的斜率$k>0, l$与两坐标轴交点分别为$A$、$B$, 求三角形$AOB$($O$为坐标原点) 面积的最小值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018841": {
+        "id": "018841",
+        "content": "已知直线$l$的方程为$y=-(a+1) x+a-2$($a \\in \\mathbf{R}$).\\\\\n(1) 若直线$l$在两坐标轴上的截距相等, 求$a$的值;\\\\\n(2) 若直线$l$不经过平面直角坐标系的第二象限, 求$a$的取值范围.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018842": {
+        "id": "018842",
+        "content": "已知直线$l$经过点$A(2,1)$、$B(4,4)$, 求直线$l$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018843": {
+        "id": "018843",
+        "content": "如图, $\\triangle ABC$三个顶点的坐标分别为$A(-2,-2)$、$B(5,0)$、$C(-2,3)$, 分别求边$AC$所在直线的方程与边$AB$上的中线$CD$所在直线的方程.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-3,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,-2) node [below] {$A$} coordinate (A);\n\\draw (-2,3) node [above] {$C$} coordinate (C);\n\\draw (5,0) node [below] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--cycle(C)--(D);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018844": {
+        "id": "018844",
+        "content": "已知直线$l$与$x$轴、$y$轴分别交于$A$、$B$两个不同的点且$|OA|=|OB|$, 其中$O$是坐标原点; 又点$C(1,-2)$在直线$l$上. 求直线$l$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018845": {
+        "id": "018845",
+        "content": "已知直线$l$经过点$(3,0)$和点$(-3,4)$. 若点$(x, y)$在直线$l$上移动且在第一象限内, 求$x y$的最大值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018846": {
+        "id": "018846",
+        "content": "已知$t \\in \\mathbf{R}$, 点$P$的坐标为$(t, t)$, 点$Q$的坐标为$(10-t, 0)$. 若直线$PQ$经过点$M(5,1)$, 求直线$PQ$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018847": {
+        "id": "018847",
+        "content": "根据下列条件, 写出直线的方程.\\\\\n(1) 过点$P(-1,2)$且斜率为$2$的直线方程为\\blank{50};\\\\\n(2) 斜率为$1$且在$y$轴上的截距为$2$的直线方程为\\blank{50};\\\\\n(3) 过点$(-1,1)$ 、$(1,5)$的直线方程为\\blank{50};\\\\\n(4) 过点$(1,2)$且与$x$轴平行的直线方程为\\blank{50};\\\\\n(5) 过点$(1,2)$且与$y$轴平行的直线方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018848": {
+        "id": "018848",
+        "content": "在平面直角坐标系中, 根据所给直线方程, 作出相应的图形, 并求出该直线的斜率和在$y$轴上的截距:\\\\\n(1) $l_1: 2 y+1=0$;\\\\\n(2) $l_2: x+2 y+1=0$.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018849": {
+        "id": "018849",
+        "content": "已知直线$l$的方程为$a x+y+a+1=0$($a \\in \\mathbf{R}$), 求证: 无论$a$取何值时, 直线$l$都经过一个定点, 并写出该定点的坐标.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018850": {
+        "id": "018850",
+        "content": "如图, 直线$l: (m+1) x+m y+2-m=0$经过平面直角坐标系的第一、第二与第四象限. 求实数$m$的取值范围.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.25]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,3) coordinate (S) (2,0) coordinate (T);\n\\draw ($(S)!-0.3!(T)$) -- ($(T)!-0.3!(S)$);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018851": {
+        "id": "018851",
+        "content": "根据下列条件, 写出直线的方程, 并把它化为一般式方程:\\\\\n(1) 经过点$A(8,-2)$, 斜率是$-\\dfrac{1}{2}$;\\\\\n(2) 经过点$B(4,2)$, 平行于$x$轴;\\\\\n(3) 经过点$P_1(3,-2)$, $P_2(5,-4)$.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018852": {
+        "id": "018852",
+        "content": "设直线$l$的方程为$(m+2) x+(m-1) y-2 m+3=0$($m \\in \\mathbf{R}$).\\\\\n(1) 求证: 对任何实数$m$, 直线$l$都必过第二象限;\\\\\n(2) 若直线$l$不过第三象限, 求实数$m$的取值范围.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018853": {
+        "id": "018853",
+        "content": "请写出下列直线的一个法向量.\\\\\n(1) $x-y+2=0$;\\\\\n(2) $x-1=0$;\\\\\n(3) $y=\\dfrac{1}{3} x-2$.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018854": {
+        "id": "018854",
+        "content": "根据下列条件, 求直线$l$的方程:\\\\\n(1) 已知直线$l$在$y$轴上的截距为$1$, 且$l$的一个法向量是$\\overrightarrow {n}=(2,1)$;\\\\\n(2) 已知直线$l$经过点$(1,3)$, 且$l$上的任意向量都与向量$\\overrightarrow {a}=(2,-1)$平行.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018855": {
+        "id": "018855",
+        "content": "已知$\\triangle ABC$的三个顶点的坐标分别是$A(1,6)$、$B(-1,-2)$与$C(6,3)$, 求边$BC$上的高$AD$所在直线的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018856": {
+        "id": "018856",
+        "content": "已知直线$l$经过点$P(3,2)$, 且$\\overrightarrow {n}=(m, 1-m)$是它的一个法向量. 若$l$与坐标轴围成的三角形的面积是$2$, 求实数$m$的值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018857": {
+        "id": "018857",
+        "content": "已知$A(4,6)$、$B(-3,-1)$、$C(4,-5)$三点, 求经过点$A$且与$BC$平行的直线$l$的点法式方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018858": {
+        "id": "018858",
+        "content": "已知$\\triangle ABC$的三个顶点为$A(1,6)$、$B(-1,2)$、$C(6,3)$.\\\\\n(1) 求边$AC$的垂直平分线的方程;\\\\\n(2) 求边$AB$上的中线$CF$所在直线的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018859": {
+        "id": "018859",
+        "content": "已知矩形$ABCD$的顶点$A(3,1)$, 对角线交点$M(2,2)$, 边$AB$所在直线的一个法向量$\\overrightarrow {n}=(3,2)$. 求此矩形四条边所在直线的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018860": {
+        "id": "018860",
+        "content": "判断下列两条直线的位置关系. 若相交, 求交点坐标.\\\\\n(1) $l_1: 0.5 x-y+1=0, l_2: 3 x-6 y+8=0$;\\\\\n(2) $l_1: y=2 x+3, l_2: y=-2 x+1$.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018861": {
+        "id": "018861",
+        "content": "已知直线$l_1: m x-2 y+1=0$, $l_2: x-(m+1) y+1=0$, 求实数$m$的取值范围, 使得: (1) $l_1$与$l_2$相交;\\\\\n(2) $l_1$与$l_2$平行;\\\\\n(3) $l_1$与$l_2$重合.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018862": {
+        "id": "018862",
+        "content": "若直线$l_1: x-2 y=0$, $l_2: 3 x+y-7=0$, $l_3: m x-y+4=0$三线共点, 则实数$m$的值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018863": {
+        "id": "018863",
+        "content": "数学家欧拉$1765$年在其所著的《三角形几何学》一书中提出: 任意三角形的外心、垂心、重心在同一条直线上, 后人称这条直线为欧拉线. 已知$\\triangle ABC$的顶点$A(2,0)$、$B(0,4)$, 若其欧拉线的方程为$x-y+2=0$, 求顶点$C$的坐标.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018864": {
+        "id": "018864",
+        "content": "已知直线$l_1: x+m y-2=0$与$l_2: (m-2) x+3 m y+2 m=0$互相垂直, 求实数$m$的值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018865": {
+        "id": "018865",
+        "content": "求经过点$(4,5)$且与直线$l: y=-\\dfrac{1}{3} x+2$垂直的直线$l$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018866": {
+        "id": "018866",
+        "content": "已知直角三角形$ABC$的斜边$BC$在$x$轴上, 且长度为$8$, 直角顶点$A$的坐标是$(2,4)$. 求直角边$AB$所在直线的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018867": {
+        "id": "018867",
+        "content": "已知$a \\in \\mathbf{R}$, $b>0$, 直线$(b^2+1) x+a y+2=0$与直线$x-b^2 y-1=0$互相垂直, 则$a b$的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018868": {
+        "id": "018868",
+        "content": "已知$m \\in \\mathbf{R}$, 过定点$A$的直线$x+m y=0$和过定点$B$的直线$m x-y-m+3=0$交于点$P$, 求$|PA|+|PB|$的取值范围.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018869": {
+        "id": "018869",
+        "content": "求直线$l_1: x+3 y-1=0$与$l_2: y=2 x+7$的夹角的大小.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018870": {
+        "id": "018870",
+        "content": "已知直线$l$经过点$P(2,1)$, 与直线$l: x+2 y+1=0$的夹角为$\\arccos \\dfrac{\\sqrt{5}}{5}$. 求直线$l$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018871": {
+        "id": "018871",
+        "content": "已知直线$l$与直线$l: 2 x-y+4=0$的夹角为$\\arctan \\dfrac{1}{2}$, 且它们的交点恰好在$x$轴上, 则直线$l$的方程是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018872": {
+        "id": "018872",
+        "content": "如图, 已知$\\triangle ABC$三个顶点的坐标分别为$A(1,1)$、$B(9,3)$、$C(2,5)$, $\\angle BAC$的角平分线与边$BC$交于点$D$, 求$AD$所在直线$l$的方程.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-1,0) -- (10,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,1) node [below] {$A$} coordinate (A);\n\\draw (9,3) node [right] {$B$} coordinate (B);\n\\draw (2,5) node [above] {$C$} coordinate (C);\n\\draw ($(C)!{1/3}!(B)$) node [above right] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--cycle(A)--(D);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018873": {
+        "id": "018873",
+        "content": "根据下列条件, 求点$P$到直线$l$的距离$d$:\\\\\n(1) $P(1,2)$, $l: x+2 y+5=0$;\\\\\n(2) $P(-\\sqrt{3}, 3)$, $l: y=\\sqrt{3} x+3$.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018874": {
+        "id": "018874",
+        "content": "求平行直线$l_1: \\sqrt{2} x+y+1=0$与$l_2: \\sqrt{2} x+y+4=0$之间的距离.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018875": {
+        "id": "018875",
+        "content": "已知直线$l_1: y=\\sqrt{2} x+1$, 直线$l_2$与$l_1$平行, 且与$l_1$的距离为$2$. 求直线$l_2$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018876": {
+        "id": "018876",
+        "content": "在直角坐标平面上, 点$A(1,2)$到直线$l$的距离为$1$, 点$B(3,1)$到直线$l$的距离为$2$, 求直线$l$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018877": {
+        "id": "018877",
+        "content": "已知两平行直线$l_1$与$l_2$分别过点$P_1(1,0)$与$P_2(0,5)$. 若$l_1$与$l_2$之间的距离为$5$, 求直线$l_1$、$l_2$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018878": {
+        "id": "018878",
+        "content": "已知$\\triangle ABC$的三个顶点的坐标分别是点$A(5,1)$、$B(4,3)$与$C(0,-1)$, 直线$m: (k+2) x+(k-1) y+k-1=0$($k \\in \\mathbf{R}$).\\\\\n(1) 求直线$AB$的斜率与倾斜角;\\\\\n(2) 求边$AC$所在直线$l_1$的方程和边$AC$上的高所在直线$l_2$的方程;\\\\\n(3) 判断直线$m$与直线$AB$的位置关系;\\\\\n(4) 记$d$为点$A$到直线$m$的距离, 试问: $d$是否存在最大值? 若存在, 求出$d$的最大值; 若不存在, 说明理由;\\\\\n(5) 若直线$l$过点$C$, 且被两条平行直线$2 x+y-6=0$与$4 x+2 y-5=0$截得的线段长为$\\dfrac{7}{2}$, 求直线$l$的方程;\\\\\n(6) 若直线$l$过点$C$, 且点$A$与点$B$到直线$l$的距离相等, 求直线$l$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018879": {
+        "id": "018879",
+        "content": "已知直线$l$过点$P(0,-1)$, 若直线$l$分别与直线$x+y-3=0$、$x-2 y+4=0$交于点$M$、$N$, 且点$P$恰为线段$MN$的中点, 求直线$l$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018880": {
+        "id": "018880",
+        "content": "若点$P(3,-2)$、$Q(-5,4)$到直线$l$的距离都等于￥4￥, 求直线$l$的方程;",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018881": {
+        "id": "018881",
+        "content": "若点$P(3,-2)$、$Q(-5,4)$到直线$l$的距离都等于$\\lambda$, 试讨论: 对于不同的非负常数$\\lambda$, 满足条件的直线$l$有多少条?",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂选择性必修第一册平面直角坐标系中的直线例题与习题",
+        "edit": [
+            "20230707\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
     "020001": {
         "id": "020001",
         "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",