From 5b3f29f14f43ccc0f7ef4b2a40df96d07c7df931 Mon Sep 17 00:00:00 2001 From: wangweiye7840 Date: Thu, 22 Feb 2024 15:16:50 +0800 Subject: [PATCH] =?UTF-8?q?=E6=94=B6=E5=BD=95=E9=AB=98=E4=BA=8C=E4=B8=8B?= =?UTF-8?q?=E5=AD=A6=E6=9C=9F=E5=91=A8=E6=9C=AB=E5=8D=B702=E6=96=B0?= =?UTF-8?q?=E9=A2=98?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 377 ++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 363 insertions(+), 14 deletions(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 1b3c21f6..c4c6ca15 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -76146,7 +76146,9 @@ "20220625\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040968" + ], "remark": "", "space": "", "unrelated": [] @@ -107027,7 +107029,9 @@ "20220701\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040967" + ], "remark": "", "space": "", "unrelated": [] @@ -115103,7 +115107,9 @@ "same": [ "011562" ], - "related": [], + "related": [ + "040964" + ], "remark": "", "space": "", "unrelated": [] @@ -250939,7 +250945,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040959" + ], "remark": "", "space": "4em", "unrelated": [] @@ -252091,7 +252099,9 @@ "same": [ "021190" ], - "related": [], + "related": [ + "040969" + ], "remark": "", "space": "4em", "unrelated": [] @@ -257257,7 +257267,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040962" + ], "remark": "", "space": "4em", "unrelated": [] @@ -324358,7 +324370,9 @@ "same": [ "003678" ], - "related": [], + "related": [ + "040964" + ], "remark": "", "space": "", "unrelated": [] @@ -369873,7 +369887,9 @@ "20230118\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040965" + ], "remark": "", "space": "", "unrelated": [] @@ -370179,7 +370195,9 @@ "20230118\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040958" + ], "remark": "", "space": "", "unrelated": [] @@ -405169,7 +405187,9 @@ "20230209\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040965" + ], "remark": "", "space": "", "unrelated": [] @@ -420821,7 +420841,9 @@ "20230413\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040965" + ], "remark": "", "space": "", "unrelated": [] @@ -575128,7 +575150,9 @@ "same": [ "008890" ], - "related": [], + "related": [ + "040969" + ], "remark": "", "space": "4em", "unrelated": [] @@ -707171,7 +707195,8 @@ "related": [ "010685", "008930", - "000282" + "000282", + "040959" ], "remark": "因答案能明显地反映出思考过程, 故从282改为填空题, 内容不变", "space": "", @@ -727445,7 +727470,9 @@ "20230302\t王伟叶" ], "same": [], - "related": [], + "related": [ + "040967" + ], "remark": "", "space": "", "unrelated": [] @@ -747801,5 +747828,327 @@ "remark": "", "space": "4em", "unrelated": [] + }, + "040956": { + "id": "040956", + "content": "已知椭圆的中心在坐标原点, 对称轴为坐标轴, 且椭圆上点到坐标原点距离的最大值为 5 , 最小值为 4 , 则此椭圆的焦距为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040957": { + "id": "040957", + "content": "已知直线 $y=x+m$ 与椭圆 $\\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$ 相交, 则实数 $m$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040958": { + "id": "040958", + "content": "直线 $y=x+1$ 被椭圆 $\\dfrac{x^2}{3}+y^2=1$ 所截得的弦长等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [ + "013136" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040959": { + "id": "040959", + "content": "若直线 $y=k x+2$ 与椭圆 $2 x^2+3 y^2=6$ 只有一个公共点, 则实数 $k$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [ + "031389", + "008852" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040960": { + "id": "040960", + "content": "已知椭圆 $3 x^2+4 y^2=12$ 和直线 $l: y=x+9$, 在直线 $l$ 上取一点 $M$, 经过点 $M$ 且以已知椭圆的焦点 $F_1$、$F_2$ 为焦点作另一个椭圆, 为使所作椭圆的长轴最短, 点 $M$ 的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040961": { + "id": "040961", + "content": "若椭圆 $\\dfrac{x^2}{m}+\\dfrac{y^2}{n}=1$ 与直线 $l: x+y-1=0$ 交于 $A, B$ 两点, $M$ 是 $A, B$ 的中点, 若 $k_{OM}=\\dfrac{\\sqrt{2}}{2}$, 则 $\\dfrac{n}{m}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040962": { + "id": "040962", + "content": "若点 $P(6,2)$ 是椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$ 外一点, 则过点 $P$ 的直线被椭圆所截得的弦中点 $Q$ 的轨迹方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [ + "009087" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040963": { + "id": "040963", + "content": "垂直于直线 $x+y=0$ 的直线 $l$ 交椭圆 $x^2+\\dfrac{y^2}{4}=1$ 于两点 $M, N$, 且 $|MN|=2$, 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": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040964": { + "id": "040964", + "content": "若双曲线 $\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$ 的两焦点为 $F_1, F_2, A$ 是该双曲线上的一点, 且 $|AF_1|=7$, 则 $|AF_2|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [ + "003678", + "011562" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040965": { + "id": "040965", + "content": "双曲线的方程为 $\\dfrac{y^2}{16}-\\dfrac{x^2}{9}=1$, 则它的焦点坐标为\\blank{50}, 渐近线为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [ + "013126", + "014469", + "015040" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "040966": { + "id": "040966", + "content": "渐近线方程为 $y= \\pm \\dfrac{3}{4}x$, 焦距为 10 的双曲线的标准方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "040967": { + "id": "040967", + "content": "已知 $P$ 是椭圆 $\\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$ 上的一点, $F_1$、$F_2$ 是其两个焦点.\\\\\n(1) 求 $|PF_1| \\cdot|PF_2|$ 的最大值;\\\\\n(2) 求 $\\overrightarrow{PF_1}\\cdot \\overrightarrow{PF_2}$ 的取值范围;\\\\\n(3) 若 $\\angle F_1PF_2=60^{\\circ}$, 求 $\\triangle F_1PF_2$ 的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [ + "040152", + "003402" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "040968": { + "id": "040968", + "content": "(1) 已知点 $P$ 在圆 $C: x^2+(y-4)^2=1$ 上移动, 点 $Q$ 在椭圆 $\\dfrac{x^2}{4}+y^2=1$ 上移动, 求 $|PQ|$ 的最大值;\\\\\n(2) 已知点 $P(0, t)$($t>0$), 点 $Q$ 在椭圆 $\\dfrac{x^2}{4}+y^2=1$ 上移动, 求 $|PQ|$ 的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [ + "002345" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "040969": { + "id": "040969", + "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 . 设直线 $l: y=x+2$与椭圆 $C$ 交于 $A$、$B$ 两点, 求:\\\\\n(1) 线段 $AB$ 中点的坐标;\\\\\n(2) $\\triangle OAB$ 的面积 ($O$ 是坐标原点).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [ + "021190", + "008890" + ], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "040970": { + "id": "040970", + "content": "已知中心在原点 $O$, 左焦点为 $F_1(-1,0)$ 的椭圆 $C_1$ 的左顶点为 $A$, 上顶点为 $B, F_1$ 到直线 $AB$ 的距离为 $\\dfrac{\\sqrt{7}}{7}|OB|$.\\\\\n(1) 求椭圆 $C_1$ 的方程;\\\\\n(2) 过点 $P(3,0)$ 作直线 $l$, 使其交椭圆 $C_1$ 于 $R$、$S$ 两点, 交直线 $x=1$ 于 $Q$ 点. 问: 是否存在这样的直线 $l$, 使 $|PQ|$ 是 $|PR|$、$|PS|$ 的等比中项? 若存在, 求出直线 $l$ 的方程; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20240222\t蔡海鸾" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] } } \ No newline at end of file