From ef2f807dd02ae598adaffaa304625607ab61ad94 Mon Sep 17 00:00:00 2001 From: wangweiye7840 Date: Fri, 7 Jul 2023 13:53:46 +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=E5=9C=86=E9=94=A5=E6=9B=B2?= =?UTF-8?q?=E7=BA=BF=E4=BE=8B=E9=A2=98=E4=B8=8E=E4=B9=A0=E9=A2=98=E5=B9=B6?= =?UTF-8?q?=E5=AF=B9=E5=BA=94=E5=8D=95=E5=85=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 2728 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 2728 insertions(+) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 292bf08d..64990b87 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -485081,6 +485081,2734 @@ "space": "4em", "unrelated": [] }, + "018882": { + "id": "018882", + "content": "已知两点$P(3,4)$、$Q(-5,6)$, 求以$PQ$为直径的圆$C$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018883": { + "id": "018883", + "content": "设平面上有一条长度为$4$的线段$AB$, 试建立适当的平面直角坐标系, 求到线段$AB$两端点的距离的平方和为$16$的点的轨迹方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018884": { + "id": "018884", + "content": "造船时, 为了船体放样, 要画出甲板圆弧线. 由于这条圆弧线的半径很大, 无法直接在钢板上用圆规画出, 需要先建立这条圆弧线的方程, 再用描点汪画出圆弧线. 如图, 已知圆弧$\\overset\\frown{AB}$的半径$r$为$29 \\mathrm{m}$, 圆弧$\\overset\\frown{AB}$所对的弦长$l$为$12 \\mathrm{m}$, 以$\\mathrm{m}$为单位, 建立适当的平面直角坐标系, 并求圆弧$\\overset\\frown{AB}$的方程. (结果精确到$0.001 \\mathrm{m}$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.2]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (-12,0) node [below] {$A$} coordinate (A);\n\\draw (B) arc ({90-atan(6/29)}:{90+atan(6/29)}:29);\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018885": { + "id": "018885", + "content": "已知一个圆与$y$轴相切, 其圆心在直线$x-3 y=0$上, 且直线$y=x$被该圆截得的弦长为$2 \\sqrt{7}$. 求此圆的方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018886": { + "id": "018886", + "content": "已知$\\alpha$: ``曲线$C$上的点的坐标都是方程$F(x, y)=0$的解'', $\\beta$: ``曲线$C$是方程$F(x, y)=0$的曲线'', 则$\\beta$是$\\alpha$的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018887": { + "id": "018887", + "content": "设直角$\\triangle ABC$的斜边$BC$的长为$4$, 试建立适当的平面直角坐标系, 求顶点$A$的轨迹方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018888": { + "id": "018888", + "content": "判断下列各方程是不是圆的方程, 如果是圆的方程, 指出圆心坐标和半径.\\\\\n(1) $x^2+y^2-x=0$;\\\\\n(2) $x^2+y^2-x+y+1=0$;\\\\\n(3) $4 x^2+4 y^2-4 x+12 y+9=0$;\\\\\n(4) $x^2+y^2-2 a y+a=0(a>0)$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018889": { + "id": "018889", + "content": "求经过$A(1,0)$、$B(3,0)$、$C(2,2)$三点的圆的方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018890": { + "id": "018890", + "content": "若圆$x^2+y^2-2 x+t=0$与直线$2 x+y=0$相交于$A$、$B$两点, 且$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=-1$(其中点$O$为坐标原点). 求实数$t$的值和圆的方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018891": { + "id": "018891", + "content": "求过原点$(0,0)$和点$A(-1,3)$, 且在$x$轴上截得弦长为$2$的圆的方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018892": { + "id": "018892", + "content": "已知圆$C: x^2+y^2+6 x-2 y+6=0$, 若直线$l$过点$A(4,0)$, 且被圆$C$截得的弦长为$2 \\sqrt{3}$, 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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018893": { + "id": "018893", + "content": "已知圆$O$的方程是$x^2+y^2=9$. 当实数$b$为何值时, 直线$l: 2 x-y+b=0$与圆$O$分别有两个不同的公共点? 有一个公共点? 没有公共点?", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018894": { + "id": "018894", + "content": "(1) 如图, 已知$M(x_0, y_0)$为圆$O: x^2+y^2=4$上一点, 求过点$M$的圆$O$的切线$l$的方程;\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,0) circle (2);\n\\draw (130:2) node [above left] {$M$} coordinate (M);\n\\draw (M) --++ (40:2) node [right] {$l$};\n\\draw (M) --++ (220:2) (O)--(M);\n\\end{tikzpicture}\n\\end{center}\n(2) 求过点$N(2,2 \\sqrt{3})$且与圆$O: x^2+y^2=4$相切的直线的方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018895": { + "id": "018895", + "content": "过圆$O: x^2+y^2=16$外一点$M(2,-6)$任意作一条割线交圆$O$于$A$、$B$两点, 求弦$AB$的中点$C$的轨迹.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018896": { + "id": "018896", + "content": "已知圆$x^2+y^2=r^2$($r>0$), 若$m \\in \\mathbf{R}$时, 直线$(m+1) x-(2 m-1) y+5 m-4=0$恒与此圆有公共点, 求圆的半径$r$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018897": { + "id": "018897", + "content": "已知圆$C: x^2+y^2-2 x+4 y-4=0$, 是否存在斜率为$1$的直线$l$, 使以$l$被圆$C$截得的弦$AB$为直径的圆经过坐标原点? 若存在, 写出直线$l$的方程; 若不存在, 说明理由.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018898": { + "id": "018898", + "content": "证明圆$C_1: x^2+y^2-4 x+6 y+5=0$与圆$C_2: x^2+y^2-6 x+4 y+11=0$内切, 并求切点坐标.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018899": { + "id": "018899", + "content": "如图, 圆$O_1$与圆$O_2$的半径都是$1$, $|O_1O_2|=4$, 过动点$P$分别作圆$O_1$、 圆$O_2$的切线$PM$、$PN$($M$、$N$分别为切点), 使得$|PM|=\\sqrt{2}|PN|$. 试通过建立适当的平面直角坐标系, 求动点$P$的轨迹.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale=0.5]\n\\draw (1,{sqrt(8)}) node [above] {$P$} coordinate (P);\n\\draw ({25/9},{sqrt(32)/9}) coordinate (T) node [right] {$N$};\n\\draw ({1/17*(-31-sqrt(128))},{2/17*(6+sqrt(2))}) coordinate (S) node [left] {$M$};\n\\draw (S)--(P)--(T);\n\\filldraw (S) circle (0.03) (T) circle (0.03);\n\\draw (-2,0) circle (1) (2,0) circle (1);\n\\filldraw (-2,0) circle (0.03) node [below] {$O_1$} (2,0) circle (0.03) node [below] {$O_2$};\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018900": { + "id": "018900", + "content": "已知圆$C_1: x^2+y^2-2 a x-2 y+a^2-15=0$, 圆$C_2: x^2+y^2-4 a x-2 y+4 a^2=0 $($a>0$), 求实数$a$的值或取值范围, 分别使得圆$C_1$与圆$C_2$\\\\\n(1) 相切;\\\\\n(2) 相交;\\\\\n(3) 内含.", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018901": { + "id": "018901", + "content": "若圆$x^2+y^2=4$与圆$x^2+y^2+2 a y-6=0$($a>0$)的公共弦的长为$2 \\sqrt{3}$, 求实数$a$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018902": { + "id": "018902", + "content": "若点$P$是圆$C_1: x^2+y^2-8 x-4 y+11=0$上的动点, 点$Q$是圆$C_2: x^2+y^2+4 x+2 y-1=0$上的动点, 则$|PQ|$的取值范围是\\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018903": { + "id": "018903", + "content": "求圆心在直线$x-y-4=0$上, 且经过两圆: $x^2+y^2+6 x-4=0$和$x^2+y^2+6 y-28=0$的交点的圆方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018904": { + "id": "018904", + "content": "已知椭圆的焦点在坐标轴上且关于原点对称, 焦距是$6$, 椭圆上的一点到两个焦点的距离之和等于$10$. 求椭圆的标准方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018905": { + "id": "018905", + "content": "已知动点$M(x, y)$到点$F_1(0,-3)$、$F_2(0,3)$的距离之和为$6$, 求满足该条件的动点$M$的轨迹方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018906": { + "id": "018906", + "content": "求焦点在$x$轴上且关于原点对称, 焦距为$2 \\sqrt{6}$, 且过点$(\\sqrt{3}, \\sqrt{2})$的椭圆的标准方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018907": { + "id": "018907", + "content": "已知$P$是椭圆$\\dfrac{x^2}{100}+\\dfrac{y^2}{36}=1$上的点, $F_1$、$F_2$是椭圆的两个焦点. 若$|PF_1|=6$, 则$|PF_2|=$\\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018908": { + "id": "018908", + "content": "已知椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$, $F_1$、$F_2$是椭圆的两个焦点. 过点$F_2$的直线垂直于$x$轴, 交椭圆于$A$、$B$两点, 那么$\\triangle AF_1B$的周长是\\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018909": { + "id": "018909", + "content": "已知椭圆的焦点在$y$轴上且关于原点对称, 焦距为$2 \\sqrt{15}$, 椭圆上的一点到两个焦点的距离之和等于$8$, 求这个椭圆的标准方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018910": { + "id": "018910", + "content": "已知椭圆的两个焦点坐标分别是$(-2,0)$、$(2,0)$, 并且经过点$(\\dfrac{5}{2},-\\dfrac{3}{2})$, 求它的标准方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018911": { + "id": "018911", + "content": "已知$\\alpha \\in[0, \\dfrac{\\pi}{2}]$, 讨论方程$x^2 \\sin \\alpha+y^2 \\cos \\alpha=1$所表示的曲线.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018912": { + "id": "018912", + "content": "已知一动圆过点$A(2,0)$, 且与圆$C: (x+2)^2+y^2=36$内切, 求该动圆的圆心$P$的轨迹方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018913": { + "id": "018913", + "content": "写出下列各椭圆的长轴长、短轴长、离心率以及焦点和顶点的坐标.\\\\\n(1) $\\dfrac{x^2}{144}+\\dfrac{y^2}{169}=1$;\\\\\n(2) $9 x^2+25 y^2=225$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018914": { + "id": "018914", + "content": "用离心率作为指标衡量, 下列两个椭圆中哪个更接近圆?\\\\\n(1) $\\dfrac{x^2}{144}+\\dfrac{y^2}{169}=1$;\\\\\n(2) $9 x^2+25 y^2=225$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018915": { + "id": "018915", + "content": "分别求适合下列条件的椭圆的标准方程:\\\\\n(1) 已知椭圆以原点为中心, 焦点在$x$轴上, 长半轴的长为$6$, 离心率为$\\dfrac{1}{3}$;\\\\\n(2) 已知椭圆以原点为中心, 焦点在坐标轴上, 长轴长为$4$, 两焦点与短轴的一个端点构成一个等边三角形;\\\\\n(3) 已知椭圆以原点为中心, 焦点在坐标轴上, 且经过$P(-3,0)$、$Q(0,-2)$两点.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018916": { + "id": "018916", + "content": "用离心率作为指标衡量, 椭圆$2 x^2+y^2=36$与椭圆$\\dfrac{3 y^2}{16}+\\dfrac{x^2}{12}=1$哪一个更接近圆?", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018917": { + "id": "018917", + "content": "已知下列椭圆的方程, 分别求椭圆的长轴长、短轴长、离心率、焦点坐标和顶点坐标:\\\\\n(1) $x^2+4 y^2=16$;\\\\\n(2) $9 x^2+y^2=81$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018918": { + "id": "018918", + "content": "求以原点为中心, 焦点在坐标轴上, 且经过$P(-\\sqrt{5}, 0)$、$Q(0,2 \\sqrt{2})$两点的椭圆的标准方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018919": { + "id": "018919", + "content": "求以原点为中心, 焦点在坐标轴上, 长轴长是短轴长的$3$倍, 且经过点$P(3,0)$的椭圆的标准方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018920": { + "id": "018920", + "content": "已知椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的顶点$B(0,-b)$, 点$P$在椭圆上, 求$|BP|$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018921": { + "id": "018921", + "content": "已知点$P$是椭圆$C: \\dfrac{x^2}{9}+y^2=1$上的一个动点, 点$Q$是圆$E: x^2+(y-4)^2=3$上的一个动点, 求$|PQ|$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018922": { + "id": "018922", + "content": "我国发射的第一颗人造地球卫星, 它的运行轨道是以地球的中心$F_2$为一个焦点的椭圆, 椭圆长轴的两个端点$A$、$B$分别为近地点和远地点, 卫星在近地点$A$与地球表面的距离约为$440 \\text{km}$, 在远地点$B$与地球表面的距离约为$2384 \\text{km}$, 地球中心与$A$、$B$在同一直线上. 已知地球的半径$R$约为$6371 \\text{km}$. 以$\\text{km}$为单位, 建立适当的平面直角坐标系, 求卫星轨道的方程. (结果精确到$1 \\text{km}$)", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018923": { + "id": "018923", + "content": "设直线与椭圆的方程分别为$2 x-y=b$与$\\dfrac{x^2}{75}+\\dfrac{y^2}{25}=1$, 当$b$为何值时, 它们满足下列关系:\\\\\n(1) 直线与椭圆有且只有一个公共点;\\\\\n(2) 直线与椭圆有两个不同的公共点;\\\\\n(3) 直线与椭圆没有公共点.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018924": { + "id": "018924", + "content": "已知直线$y=m x+2$与椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$有两个不同的公共点, 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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018925": { + "id": "018925", + "content": "已知直线$y=m x+1$与椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$有两个不同的公共点, 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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018926": { + "id": "018926", + "content": "过点$M(0,1)$且倾斜角为$\\dfrac{\\pi}{3}$的直线$l$与椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$是否相交于不同的两点? 若是, 求交点$A$、$B$间的距离$|AB|$; 若不是, 请说明理由.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018927": { + "id": "018927", + "content": "已知$P$是椭圆$\\dfrac{x^2}{5}+\\dfrac{y^2}{4}=1$上的点, $F_1$、$F_2$是该椭圆的两个焦点. 若三角形$PF_1F_2$的面积等于$1$, 则点$P$的坐标为\\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018928": { + "id": "018928", + "content": "已知$P$是椭圆$\\dfrac{x^2}{12}+\\dfrac{y^2}{3}=1$上的点, $F_1$、$F_2$是该椭圆的两个焦点. 若线段$PF_1$的中点$H$在$y$轴上, 则$\\dfrac{|PF_1|}{|PF_2|}=$\\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018929": { + "id": "018929", + "content": "已知直线$l: y=-x+1$与椭圆$C: m x^2+n y^2=1(m>0, n>0)$相交于$A$、$B$两点. 若$|AB|=2 \\sqrt{2}$, 且$AB$的中点$M$与椭圆中心$O$($O$为坐标原点) 的连线的斜率为$\\dfrac{\\sqrt{2}}{2}$, 求椭圆$C$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018930": { + "id": "018930", + "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+y^2=1(a>1)$的左、右焦点分别是$F_1$、$F_2$, 点$P$是椭圆$C$上的一点且在第一象限, $\\triangle PF_1F_2$的周长为$4+2 \\sqrt{3}$. 过点$P$作椭圆$C$的切线$l$, 分别与$x$轴和$y$轴交于$A$、$B$两点, $O$为坐标原点. 当点$P$在椭圆$C$上移动时, $\\triangle AOB$面积的最小值为\\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018931": { + "id": "018931", + "content": "已知双曲线的焦点在坐标轴上且关于原点对称, 焦距是$6$, 双曲线上的点到两个焦点距离之差的绝对值等于$4$. 求双曲线的标准方程和焦点的坐标.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018932": { + "id": "018932", + "content": "已知点$M(x, y)$到点$F_1(-3,0)$的距离减去它到点$F_2(3,0)$的距离之差是$m$, 分别求下列条件下点$M$的轨迹方程:\\\\\n(1) $m=4$;\\\\\n(2) $m=6$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018933": { + "id": "018933", + "content": "在相距$2000 \\mathrm{m}$的两个观察站$A$、$B$先后听到远处传来的爆炸声, 已知$A$站听到的时间比$B$站早$4 \\mathrm{s}$, 声速是$340 \\mathrm{m} / \\mathrm{s}$. 建立适当的平面直角坐标系, 判断爆炸点可能分布在什么样的轨迹上, 并求该轨迹的方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018934": { + "id": "018934", + "content": "若双曲线$2 x^2-y^2=m$的一个焦点为$(0, \\sqrt{3})$, 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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018935": { + "id": "018935", + "content": "已知$F_1$、$F_2$是双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的两个焦点, 点$P$在该双曲线上且满足$|PF_1| \\cdot|PF_2|=64$, 那么$\\angle F_1PF_2=$\\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018936": { + "id": "018936", + "content": "已知动圆$P$过点$A(3,0)$, 且与圆$(x+3)^2+y^2=4$相外切, 求动圆圆心$P$的轨迹方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018937": { + "id": "018937", + "content": "某团队在距$O$地$20 \\text{km}$的西侧、东侧分别设置$A$、$B$两个地面站点, 在距$O$地$15 \\text{km}$的南侧、北侧分别设置$C$、$D$两个地面站点, 在测量距离后发现$Q$地满足$|QA|-|QB|=30 \\text{km}$, $|QC|-|QD|=10 \\text{km}$. 若需从$O$地出发前往$Q$地, 请求出$O$、$Q$两地之间的距离(结果精确到$1 \\text{km}$) 以及行走的方向(结果精确到$1^{\\circ}$).", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018938": { + "id": "018938", + "content": "求双曲线$16 x^2-9 y^2=144$的顶点坐标、焦点坐标、离心率与渐近线方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018939": { + "id": "018939", + "content": "求双曲线$4 x^2-9 y^2=-4$的顶点坐标、焦点坐标、实轴长、虚轴长、离心率和渐近线方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018940": { + "id": "018940", + "content": "对于双曲线$C_1: \\dfrac{x^2}{16}-\\dfrac{y^2}{9}=1$和$C_2: \\dfrac{y^2}{9}-\\dfrac{x^2}{16}=1$, 给出下列四个结论: \\textcircled{1} 离心率相等; \\textcircled{2} 渐近线相同; \\textcircled{3} 没有公共点; \\textcircled{4} 焦距相等, 其中正确的结论是 \\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{2}\\textcircled{3}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018941": { + "id": "018941", + "content": "已知双曲线$E$的渐近线方程为$y= \\pm 2 x$, 则其离心率为\\bracket{20}.\n\\fourch{$\\sqrt{5}$}{$\\dfrac{\\sqrt{5}}{2}$}{$2$}{$\\dfrac{\\sqrt{5}}{2}$或$\\sqrt{5}$}", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018942": { + "id": "018942", + "content": "设双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点分别为$F_1$、$F_2$, 其渐近线方程为$y= \\pm \\dfrac{4}{3} x$.\\\\\n(1) 如果双曲线$C$的焦点都在圆$x^2+y^2=100$上, 求双曲线$C$的方程;\\\\\n(2) 如果双曲线$C$的实轴长为$6$, 点$P$为$C$的右支上一点, 且$|PF_2|=|F_1F_2|$, 求$\\triangle PF_1F_2$的面积.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018943": { + "id": "018943", + "content": "已知双曲线过点$P(4,3)$, 它的一条渐近线的方程为$y=\\dfrac{1}{2} x$. 求双曲线的标准方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018944": { + "id": "018944", + "content": "设直线与双曲线的方程分别为$y=k x$和$x^2-y^2=1$, 当实数$k$取何值时, 直线与双曲线分别有两个不同的公共点? 有且只有一个公共点? 没有公共点?", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018945": { + "id": "018945", + "content": "双曲线型自然冷却通风塔的外形是由双曲线的一部分绕其虚轴所在的直线旋转一周所形成的曲面, 如图所示. 已知它的最小半径为$12 \\mathrm{m}$, 上口半径为$13 \\mathrm{m}$, 下口半径为$25 \\mathrm{m}$, 高为$55 \\mathrm{m}$. 建立适当的平面直角坐标系, 求此双曲线的方程. (结果精确到$0.1 \\mathrm{m}$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.05, x = {(-20:1cm)}, y = {(-160:1cm)}, z = {(90:1cm)}]\n\\foreach \\i in {-45,0,90,135}\n{\\draw [domain = -44.8:10.2] plot ({12*sqrt(1+\\x*\\x/24.5/24.5)*cos(\\i)},{12*sqrt(1+\\x*\\x/24.5/24.5)*sin(\\i)},\\x);};\n\\foreach \\i in {180,270}\n{\\draw [domain = -44.8:10.2, dashed] plot ({12*sqrt(1+\\x*\\x/24.5/24.5)*cos(\\i)},{12*sqrt(1+\\x*\\x/24.5/24.5)*sin(\\i)},\\x);};\n\\def\\r{12}\n\\draw [dashed] (\\r,0,0) -- (-\\r,0,0) (0,\\r,0) -- (0,-\\r,0);\n\\draw [domain = -45:135] plot ({\\r*cos(\\x)},{\\r*sin(\\x)},0);\n\\draw [domain = 135:315, dashed] plot ({\\r*cos(\\x)},{\\r*sin(\\x)},0);\n\\def\\r{12*sqrt(1+10.2*10.2/24.5/24.5)}\n\\draw [dashed] ({12*sqrt(1+10.2*10.2/24.5/24.5)},0,10.2) -- ({-12*sqrt(1+10.2*10.2/24.5/24.5)},0,10.2) (0,{12*sqrt(1+10.2*10.2/24.5/24.5)},10.2) -- (0,{-12*sqrt(1+10.2*10.2/24.5/24.5)},10.2);\n\\draw [domain = -45:135] plot ({\\r*cos(\\x)},{\\r*sin(\\x)},10.2);\n\\draw [domain = 135:315] plot ({\\r*cos(\\x)},{\\r*sin(\\x)},10.2);\n\\def\\r{12*sqrt(1+44.8*44.8/24.5/24.5)}\n\\draw [dashed] ({12*sqrt(1+44.8*44.8/24.5/24.5)},0,-44.8) -- ({-12*sqrt(1+44.8*44.8/24.5/24.5)},0,-44.8) (0,{12*sqrt(1+44.8*44.8/24.5/24.5)},-44.8) -- (0,{-12*sqrt(1+44.8*44.8/24.5/24.5)},-44.8);\n\\draw [domain = -45:135] plot ({\\r*cos(\\x)},{\\r*sin(\\x)},-44.8);\n\\draw [domain = 135:315, dashed] plot ({\\r*cos(\\x)},{\\r*sin(\\x)},-44.8);\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018946": { + "id": "018946", + "content": "如图, 某绿色蔬菜种植基地在$A$处, 要把此处生产的蔬菜沿道路$AA_1$或$AA_2$运送到四边形区域$A_1A_2A_3A_4$的农贸市场. 现要求在农贸市场中确定一条界线, 使位于界线一侧的点沿道路$AA_1$比沿道路$AA_2$运送蔬菜近, 而另一侧的点则反之, 该界线所在曲线为\\bracket{20}.\n\\fourch{直线}{椭圆}{双曲线}{抛物线}\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (2,0) node [right] {$A_2$} coordinate (A_2);\n\\draw (0.5,-1.5) node [below] {$A$} coordinate (A);\n\\draw (2.5,1) node [right] {$A_3$} coordinate (A_3);\n\\draw (0.3,1.8) node [above] {$A_4$} coordinate (A_4);\n\\filldraw [pattern = north east lines] (A_1)--(A_2)--(A_3)--(A_4)--cycle;\n\\draw (A_1)--(A)--(A_2);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018947": { + "id": "018947", + "content": "已知动点$P$在双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)上, 双曲线的左顶点为$A$, 右焦点为$F$, 且$PF \\perp AF$, $2|PF|=|AF|$, 求该双曲线的渐近线方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018948": { + "id": "018948", + "content": "已知双曲线$C: \\dfrac{x^2}{4}-\\dfrac{y^2}{3}=1$.\\\\\n(1) 求与双曲线$C: \\dfrac{x^2}{4}-\\dfrac{y^2}{3}=1$共渐近线且过点$M(2,2 \\sqrt{3})$的双曲线的方程;\\\\\n(2) 若斜率为$k$的直线$l$过点$Q(-1,0)$, 且与双曲线$C$的左、右支各有一个公共点, 求$k$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018949": { + "id": "018949", + "content": "某团队开发一款``猫捉老鼠''的游戏. 如图所示, $A$、$B$两个信号源相距$10$米, $O$是$AB$的中点, 过点$O$的直线$l$与直线$AB$的夹角为$45^{\\circ}$. 机器猫在直\n线$l$上运动, 机器鼠的运动轨迹始终满足: 接收到点$A$的信号比接收到点$B$的信号晩$\\dfrac{8}{v_0}$秒, 其中$v_0$(单位: 米/秒) 是信号传播的速度. 游戏设定: 机器鼠在距离直线$l$不超过 $1.5$米的区域运动时, 有``被抓''的风险, 如果机器鼠保持目前的运动轨迹不变, 是否有``被抓''的风险?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\filldraw (-1,0) circle (0.03) node [below] {$A$} coordinate (A);\n\\filldraw (1,0) circle (0.03) node [below] {$B$} coordinate (B);\n\\draw (-1.6,-1.6) -- (1.6,1.6) node [right] {$l$};\n\\draw (0.8,0.8) node [fill = white] {\\rotatebox{45}{猫}};\n\\draw ({4/3},0.8) node {鼠};\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018950": { + "id": "018950", + "content": "求顶点在坐标原点, 焦点在坐标轴上且经过点$M(-2,-4)$的抛物线的方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018951": { + "id": "018951", + "content": "证明: 以抛物线$y^2=2 p x$的任一过焦点的弦为直径的圆与抛物线的准线相切.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018952": { + "id": "018952", + "content": "抛物线$C: y^2=4 x$上一点$N$到点$Q(7,8)$与到$C$的准线的距离之和的最小值为\\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018953": { + "id": "018953", + "content": "已知$A(3,1)$, $F$是抛物线$y^2=4 x$的焦点, $P$是抛物线上的一个动点, 求$\\triangle APF$周长的最小值.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018954": { + "id": "018954", + "content": "已知抛物线$C: y^2=2 p x$($p>0$)的焦点为$F$, 点$M$是抛物线$C$上一点, 圆$M$与$y$轴相切且被直线$x=\\dfrac{p}{2}$截得的弦长为$\\sqrt{2} p$, 若$|MF|=\\dfrac{5}{2}$, 求抛物线$C$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018955": { + "id": "018955", + "content": "求过定点$M(0,1)$且与抛物线$y^2=2 x$只有一个公共点的直线的方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018956": { + "id": "018956", + "content": "在抛物线$y^2=2 x$上求一点$M$, 使点$M$到直线$l: y=\\dfrac{1}{2} x+3$的距离最短, 并求最短距离.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018957": { + "id": "018957", + "content": "过抛物线$y^2=4 x$的焦点且斜率为$2$的直线与抛物线相交于$A$、$B$两点, 求线段$AB$的长.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018958": { + "id": "018958", + "content": "如图, 汽车前灯反射镜与轴截面的交线是抛物线的一部分, 灯泡位于抛物线的焦点处. 已知灯口直径是$24 \\text{cm}$, 灯深$10 \\text{cm}$, 求灯泡与反射镜顶点的距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.15]\n\\filldraw (3.6, 0) circle (0.1) node [right] {$F$} coordinate (F);\n\\filldraw (10, 0) circle (0.1);\n\\draw [domain = -12:12] plot ({pow(\\x, 2)/14.4}, \\x);\n\\draw (10, 0) ellipse (3 and 12);\n\\draw (0, 0) node [left] {$O$} coordinate (O) --++ (0, -14);\n\\draw (10, -12) --++ (0, -2);\n\\draw [<->] (0, -13) -- (10, -13) node [midway, below] {$10\\text{cm}$};\n\\draw (10, -12) --++ (5, 0) (10, 12) --++ (5, 0);\n\\draw [<->] (14, -12) -- (14, 12) node [midway, right] {\\rotatebox{90}{$24\\text{cm}$}};\n\\draw [dashed] (10, 0) --++ (0, -12);\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018959": { + "id": "018959", + "content": "若抛物线$y^2=2 x$上的$A$、$B$两点与坐标原点$O$构成正三角形, 则该正三角形的边长为\\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018960": { + "id": "018960", + "content": "如图, 长度为$7$的线段$AB$的两个端点在抛物线$x^2=4 y$上运动, 求线段$AB$的中点$G$到$x$轴的距离的最小值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.25]\n\\draw [->] (-6,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,9) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -6:6] plot (\\x,{\\x*\\x/4});\n\\filldraw (0,1) circle (0.12) node [right] {$F$} coordinate (F);\n\\draw (-2,1) node [left] {$B$} coordinate (B);\n\\draw (4.16,4.33) node [right] {$A$} coordinate (A);\n\\draw (A)--(B);\n\\draw ($(A)!0.5!(B)$) node [above] {$G$} coordinate (G);\n\\filldraw (G) circle (0.12);\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018961": { + "id": "018961", + "content": "在平面直角坐标系$x O y$中, 若直线$y=k x+1$与抛物线$x^2=2 y$相交于$A$、$B$两点.\\\\\n(1) 求$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的值;\\\\\n(2) 若$\\triangle AOB$的面积为$2$, 求实数$k$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018962": { + "id": "018962", + "content": "(1) 方程$x=\\sqrt{1-y^2}$是圆心在坐标原点、半径为$1$的圆的方程吗? 为什么?\n(2) 方程$x^2-y^2=0$是过点$(0,0)$与$(1,1)$的直线的方程吗? 为什么?", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018963": { + "id": "018963", + "content": "已知点$A$、$B$是距离为$4$的两个定点, 动点$M$满足$\\overrightarrow{MA} \\cdot \\overrightarrow{MB}=5$. 建立适当的平面直角坐标系, 求动点$M$的轨迹方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018964": { + "id": "018964", + "content": "求连接定点$A(4,0)$和曲线$x^2+2 y^2=1$上动点$B$的线段$AB$的中点$P$的轨迹方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018965": { + "id": "018965", + "content": "设两条直线$l_1: a_1 x+b_1 y+1=0$和$l_2: a_2 x+b_2 y+1=0$交于点$(2,3)$, 则过点$P(a_1, b_1)$和$Q(a_2, b_2)$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018966": { + "id": "018966", + "content": "在平面直角坐标系$x O y$中, $O$为坐标原点, 定点$A(-2,3)$, 动点$B$在曲线$x^2+4 y^2=4$上运动, 以$OA$、$OB$为两边作平行四边形$OACB$, 求动点$C$的轨迹方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018967": { + "id": "018967", + "content": "求所有斜率为$1$的直线被椭圆$\\dfrac{x^2}{4}+y^2=1$所截得的线段的中点的轨迹.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018968": { + "id": "018968", + "content": "已知点$M(x, y)$在椭圆$C: \\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$上, 求$x+y$的最大值, 并求$x+y$取得最大值时点$M$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018969": { + "id": "018969", + "content": "直线$l$的参数方程为$\\begin{cases}x=1+\\sqrt{3} t, \\\\ y=1-3 t\\end{cases}$($t \\in \\mathbf{R}$), 则直线$l$的倾斜角为\\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018970": { + "id": "018970", + "content": "在平面直角坐标系$x O y$中, 曲线$C_1$的一个参数方程为$\\begin{cases}x=2 \\sin \\theta, \\\\ y=2 \\cos \\theta\\end{cases}$($\\theta \\in \\mathbf{R}$).\\\\\n(1) 写出曲线$C_1$的一个普通方程;\\\\\n(2) 设点$M(1,0)$, 点$N$是曲线$C_1$上的动点, 动点$P$满足$\\overrightarrow{OP}=2 \\overrightarrow{OM}+\\overrightarrow{ON}$, 写出点$P$运动轨迹的一个参数方程和一个普通方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018971": { + "id": "018971", + "content": "已知点$M(x, y)$在椭圆$C: \\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$上, 尝试用不同的方法求$x+y^2$的最大值, 并求$x+y^2$取得最大值时点$M$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018972": { + "id": "018972", + "content": "已知: $\\begin{cases}x=1+r \\cos \\theta, \\\\y=2+r \\sin \\theta.\\end{cases}$\\\\\n(1) 若$\\theta$为参数, $r$为常数($\\theta \\in \\mathbf{R}$, $r>0$), 则该参数方程所确定的曲线$C$是什么图形?\\\\\n(2) 若$r$为参数($r \\in \\mathbf{R}$), 设$\\theta$为常数且$\\theta \\neq \\dfrac{k}{2} \\pi$($k \\in \\mathbf{Z}$), 则该参数方程所确定的曲线$C$是什么图形?", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018973": { + "id": "018973", + "content": "如图所示, 在极坐标系中, 将$2 \\pi$的极角作$24$等分, 写出$A$、$B$、$C 、D$、$E$各点的一个极坐标.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\foreach \\i in {0,15,...,345}\n{\\draw [dashed] (0,0) -- (\\i:6);};\n\\foreach \\i in {1,2,3,4,5}\n{\\draw [dashed] (0,0) circle (\\i);\n\\draw (\\i,0) node [below] {\\small$\\i$};};\n\\filldraw (30:4) circle (0.15) node [right] {$A$} coordinate (A);\n\\filldraw (75:3) circle (0.15) node [left] {$B$} coordinate (B);\n\\filldraw (210:2) circle (0.15) node [left] {$C$} coordinate (C);\n\\filldraw (-15:5) circle (0.15) node [right] {$D$} coordinate (D);\n\\filldraw (135:1) circle (0.15) node [left] {$E$} coordinate (E);\n\\draw (45:5.5) node [above right] {$\\dfrac{\\pi}{4}$};\n\\draw (90:5.5) node [above] {$\\dfrac{\\pi}{2}$};\n\\draw (135:5.5) node [above left] {$\\dfrac{3\\pi}{4}$};\n\\draw (180:5.5) node [left] {$\\pi$};\n\\draw (225:5.5) node [below left] {$\\dfrac{5\\pi}{4}$};\n\\draw (270:5.5) node [below] {$\\dfrac{3\\pi}{2}$};\n\\draw (315:5.5) node [below right] {$\\dfrac{7\\pi}{4}$};\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018974": { + "id": "018974", + "content": "已知点$A$的极坐标是$(3, \\dfrac{\\pi}{6})$, 分别求出下列各点的一个极坐标:\\\\\n(1) 点$A$关于极轴的对称点$A_1$;\\\\\n(2) 点$A$关于极点的对称点$A_2$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018975": { + "id": "018975", + "content": "边长为$a$的正六边形$OABCDE$在极坐标系中的位置如图, 分别求这个正六边形各顶点的一个极坐标.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0) node [below] {$A$} coordinate (A);\n\\draw (A) ++ (60:1) node [right] {$B$} coordinate (B);\n\\draw (B) ++ (120:1) node [above] {$C$} coordinate (C);\n\\draw (C) ++ (180:1) node [above] {$D$} coordinate (D);\n\\draw (D) ++ (240:1) node [left] {$E$} coordinate (E);\n\\draw (O)--(A)--(B)--(C)--(D)--(E)--cycle;\n\\draw [->] (A) -- (2,0) node [below] {$x$};\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018976": { + "id": "018976", + "content": "已知点$A$的极坐标是$(3, \\dfrac{\\pi}{6})$, 分别求符合下列条件的点$A$的所有极坐标:\\\\\n(1) $\\rho>0$, $2 \\pi \\leq \\theta<4 \\pi$;\\\\\n(2) $\\rho<0$, $0 \\leq \\theta<2 \\pi$;\\\\\n(3) $\\rho<0$, $-\\pi<\\theta<\\pi$;\\\\\n(4) $\\rho \\in \\mathbf{R}$, $-2 \\pi \\leq \\theta<2 \\pi$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018977": { + "id": "018977", + "content": "在极坐标系中, 描出下列各个已知极坐标的点: $A(5, \\dfrac{\\pi}{6})$、$B(6, \\dfrac{\\pi}{4})$、$C(3, \\dfrac{\\pi}{4})$、$D(-3, \\dfrac{\\pi}{4})$、$E(5,-\\dfrac{\\pi}{6})$. 并且说明:\\\\\n(1) 点$C$和点$D$有怎样的位置关系?\\\\\n(2) 点$A$和点$E$有怎样的位置关系?\\\\\n(3)$B$、$C$、$D$三点有怎样的位置关系?", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018978": { + "id": "018978", + "content": "在极坐标系中, 已知两点$A(3,-\\dfrac{\\pi}{3})$, $B(1, \\dfrac{2 \\pi}{3})$, 求$A$、$B$两点间的距离.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018979": { + "id": "018979", + "content": "在极坐标系中, 与点$(\\rho, \\pi-\\theta)$关于极点对称的点的一个极坐标可以是\n\\fourch{$(\\rho,-\\theta)$}{$(\\rho,-\\pi-\\theta)$}{$(\\rho, \\pi-\\theta)$}{$(\\rho, \\pi+\\theta)$}", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018980": { + "id": "018980", + "content": "在极坐标系中, $O$是极点, 若$A(3, \\dfrac{\\pi}{3})$, $B(4, \\dfrac{5 \\pi}{6})$, 求$\\triangle AOB$的面积.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018981": { + "id": "018981", + "content": "在极坐标系中, $O$是极点, 已知点$M$的极坐标为$(1, \\dfrac{\\pi}{6})$, 求点$(3, \\dfrac{\\pi}{2})$关于直线$OM$对称的点的一个极坐标.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018982": { + "id": "018982", + "content": "求圆心是$C(a, 0)$、半径是$a$的圆的极坐标方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018983": { + "id": "018983", + "content": "如图, 求经过点$M(a, 0)$($a>0$), 且与极轴夹角为$\\varphi$的直线$l$的极坐标方程.\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) circle (0.03) node [left] {$O$} coordinate (O);\n\\draw [->](O) --++ (3,0) node [below] {$x$} coordinate (x);\n\\filldraw (1.3,0) circle (0.03) node [below] {$M$} coordinate (M);\n\\draw (M) --++ (1,2) node [right] {$l$} coordinate (l) (M) --++ (-0.5,-1);\n\\draw pic [draw, \"$\\varphi$\", scale = 0.5, angle eccentricity = 1.8] {angle = x--M--l};\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018984": { + "id": "018984", + "content": "设质点$M$为射线$OA$上的动点, 沿着向量$\\overrightarrow{OA}$方向做匀速运动, 同时射线$OA$又绕着它的端点$O$作等角速度旋转. 求质点$M$运动的轨迹方程.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\filldraw (0,0) circle (0.03) node [left] {$O$} coordinate (O);\n\\draw (O) --++ (3,0) coordinate (x);\n\\draw (60:2) node [right] {$A$} coordinate (A) -- (O);\n\\draw pic [draw, scale = 0.2] {angle = x--O--A};\n\\draw [domain = 0:380, samples = 300] plot (\\x:{0.5+\\x/200});\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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018985": { + "id": "018985", + "content": "已知点$A$的极坐标为$(3, \\dfrac{\\pi}{6})$, 求过点$A$且垂直于极轴的直线的极坐标方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018986": { + "id": "018986", + "content": "已知点$B$的极坐标为$(-5, \\dfrac{\\pi}{2})$, 求以点$B$为圆心且与极轴所在直线相切的圆的极坐标方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018987": { + "id": "018987", + "content": "求圆心为$C(3, \\dfrac{\\pi}{6})$、半径为$3$的圆的极坐标方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018988": { + "id": "018988", + "content": "写出下列极坐标方程表示的图形的名称:\\\\\n(1) $\\theta=\\dfrac{\\pi}{6}(\\rho \\geq 0)$;\\\\\n(2) $\\tan \\theta=1$;\\\\\n(3) $\\rho=3$;\\\\\n(4) $\\rho=-3$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018989": { + "id": "018989", + "content": "如图, 在极坐标系中, 已知直线$l$过点$A(1,0)$, 且其向上的方向与极轴的正方向所成的最小正角为$\\dfrac{\\pi}{3}$, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) circle (0.03) node [left] {$O$} coordinate (O);\n\\draw [->](O) --++ (3,0) node [below] {$x$} coordinate (x);\n\\filldraw (1,0) circle (0.03) node [below right] {$A(1,0)$} coordinate (M);\n\\draw (M) --++ (1,2) node [right] {$l$} coordinate (l) (M) --++ (-0.5,-1);\n\\draw pic [draw, \"$\\dfrac{\\pi}{3}$\", scale = 0.5, angle eccentricity = 2.8] {angle = x--M--l};\n\\end{tikzpicture}\n\\end{center}\n(1) 直线$l$的极坐标方程;\\\\\n(2) 极点到直线$l$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018990": { + "id": "018990", + "content": "已知圆$C$的圆心的极坐标为$(6, \\dfrac{\\pi}{2})$, 半径为$5$. 直线$\\theta=\\alpha$($\\rho \\in \\mathbf{R}$)被圆$C$截得的弦长为$8$, 当$\\dfrac{\\pi}{2} \\leq \\alpha<\\pi$时, 求$\\alpha$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018991": { + "id": "018991", + "content": "化直角坐标方程$x-y=0$为极坐标方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018992": { + "id": "018992", + "content": "化极坐标方程$\\rho=4 \\cos \\theta$为直角坐标方程, 并指出它是什么曲线.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018993": { + "id": "018993", + "content": "把下列各点的极坐标化成直角坐标: $A(5, \\dfrac{\\pi}{4})$, $B(3,-\\dfrac{\\pi}{6})$, $C(-4, \\dfrac{7 \\pi}{6})$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018994": { + "id": "018994", + "content": "把下列各点的直角坐标化成极坐标: $D(2,-2)$, $E(-2,2 \\sqrt{3})$, $F(0,-3)$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018995": { + "id": "018995", + "content": "把下列曲线的直角坐标方程化成极坐标方程:\\\\\n(1) $y-2=0$;\\\\\n(2) $y^2=2 x$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018996": { + "id": "018996", + "content": "把下列曲线的极坐标方程化成直角坐标方程:\\\\\n(1) $\\rho=-1$;\\\\\n(2) $4 \\theta+5 \\pi=0$;\\\\\n(2) $\\rho \\sin \\theta=2 a$;\\\\\n(4) $\\rho=-2 a \\cos \\theta$.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018997": { + "id": "018997", + "content": "求圆$\\rho=\\sqrt{2}$($\\cos \\theta+\\sin \\theta$)的圆心的一个极坐标.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "018998": { + "id": "018998", + "content": "在极坐标系中, 下列关于曲线$C: \\rho=4 \\sin (\\theta-\\dfrac{\\pi}{3})$的说法中, 正确的是\\bracket{20}.\n\\twoch{曲线$C$关于直线$\\theta=\\dfrac{5 \\pi}{6}$($\\rho \\in \\mathbf{R}$)对称}{曲线$C$关于直线$\\theta=\\dfrac{\\pi}{3}$($\\rho \\in \\mathbf{R}$)对称}{曲线$C$关于点$(2, \\dfrac{\\pi}{3})$对称}{曲线$C$关于极点$(0,0)$对称}", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "018999": { + "id": "018999", + "content": "在极坐标系中, 已知圆$\\rho=2 \\cos \\theta$与直线$3 \\rho \\cos \\theta+4 \\rho \\sin \\theta=-m$相切, 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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019000": { + "id": "019000", + "content": "若两定点$A$、$B$的距离为$3$, 动点$M$满足$|MA|=2|MB|$, 建立适当的平面直角坐标系, 求动点$M$的轨迹方程.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019001": { + "id": "019001", + "content": "椭圆$C: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$的左、右焦点分别是$F_1$、$F_2$, 点$T$是椭圆$C$上的动点.\\\\\n(1) 是否存在点$T$, 使得$\\angle F_1TF_2=90^{\\circ}$, 若存在, 求出点$T$的坐标; 若不存在, 请说明理由;\\\\\n(2) 求证: 当点$T$是椭圆短轴顶点时, $\\angle F_1TF_2$取得最大值.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019002": { + "id": "019002", + "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点分别是$F_1$、$F_2$, 在其渐近线上存在一点$P$, 满足$|PF_1|-|PF_2| |=2 b$, 求该双曲线离心率的取值范围.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019003": { + "id": "019003", + "content": "抛物线的顶点在原点, 准线方程是$x=-1$, 是否存在被点$E(1,1)$平分的弦? 如果存在, 求出弦所在直线的方程以及弦长; 如果不存在, 请说明理由.", + "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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019004": { + "id": "019004", + "content": "过抛物线$C: y^2=4 x$的焦点$F$的直线交$C$于$A$、$B$两点, 过$A$、$B$两点分别作$C$的准线的垂线, 垂足为$A_1$、$B_1$, 以线段$A_1B_1$为直径的圆$E$过点$M(-2,3)$, 求圆$E$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圆锥曲线例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019005": { + "id": "019005", + "content": "已知两圆$C_1: (x-2)^2+y^2=54$, $C_2: (x+2)^2+y^2=6$, 动圆$M$在圆$C_1$内部且和圆$C_1$内切、和圆$C_2$外切.\\\\\n(1) 求动圆圆心$M$的轨迹$C$的方程;\\\\\n(2) 过点$A(3,0)$的直线与 (1) 中的曲线$C$交于$P$、$Q$两点, 点$P$关于$x$轴对称的点为$R$, 求$\\triangle ARQ$面积的最大值.", + "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圆锥曲线例题与习题", + "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.",