收录高三寒假作业68新题

工具v2/文本文件/新题收录列表.txt

@@ -202,3 +202,6 @@
 20240125-143409
 031229,024133:024135,016576,024136
 
+20240125-143605
+024137:024142
+

题库0.3/Problems.json

@@ -649649,6 +649649,126 @@
         "space": "4em",
         "unrelated": []
     },
+    "024137": {
+        "id": "024137",
+        "content": "若双曲线 $m x^2+n y^2=1$ 的一个焦点与抛物线 $y=\\dfrac{1}{8}x^2$ 的焦点相同, 离心率为 $2$ , 则抛物线的焦点到双曲线的一条渐近线的距离为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "024138": {
+        "id": "024138",
+        "content": "若经过椭圆 $\\dfrac{x^2}{2}+y^2=1$ 的一个焦点作倾斜角为 $45^{\\circ}$ 的直线 $l$, 交椭圆于 $A$、$B$ 两点. 设 $O$为坐标原点, 则 $\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "024139": {
+        "id": "024139",
+        "content": "抛物线 $x^2=-2 y$ 与过点 $P(0,-1)$ 的直线 $l$ 交于 $A$、$B$ 两点, 若 $OA$ 与 $OB$ 的斜率之和为 $1$ , 则直线 $l$ 的方程是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "024140": {
+        "id": "024140",
+        "content": "已知动点 $P$ 在双曲线 $C: x^2-\\dfrac{y^2}{3}=1$ 上, 双曲线 $C$ 的左、右焦点分别为 $F_1$、$F_2$,\n下列结论中错误的是\\blank{50}.\\\\\n\\textcircled{1} 双曲线 $C$ 的离心率为 $\\sqrt{2}$;\\\\\n\\textcircled{2} 双曲线 $C$ 的渐近线方程为 $y= \\pm \\dfrac{\\sqrt{3}}{3}x$;\\\\\n\\textcircled{3} 双曲线 $\\dfrac{y^2}{3}-x^2=1$ 与已知双曲线 $C$ 的渐近线并不相同;\\\\\n\\textcircled{4} 动点 $P$ 到两条渐近线的距离之积为定值.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "024141": {
+        "id": "024141",
+        "content": "已知椭圆 $C: \\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1, F_1$、$F_2$ 分别为它的左、右焦点, $A$、$B$ 分别为它的左、右顶点, 已知定点 $Q(4,2)$, 点 $P$ 是椭圆上的一个动点, 下列结论中错误的是\\blank{50}.\\\\\n\\textcircled{1} 存在点 $P$, 使得 $\\angle F_1PF_2=120^{\\circ}$;\\\\\n\\textcircled{2} 直线 $PA$ 与直线 $PB$ 斜率乘积为定值;\\\\\n\\textcircled{3} $\\dfrac{1}{|PF_1|}+\\dfrac{25}{|PF_2|}$ 有最小值 $\\dfrac{18}{5}$;\\\\\n\\textcircled{4} $|PQ|+|PF_1|$ 的范围为 $[2 \\sqrt{17}, 12]$.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "024142": {
+        "id": "024142",
+        "content": "若抛物线 $C: y^2=2 p x$($p \\geq 0$) 的焦点 $F$ 与椭圆 $E: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$ 的一个焦点重合, 过坐标原点 $O$ 作两条互相垂直的射线 $OM$、$ON$, 与抛物线 $C$ 分别交于 $M$、$N$ 两点, 求证: 直线 $MN$ 恒过定点, 并求出该定点坐标.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
     "030001": {
         "id": "030001",
         "content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",