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            • 1.

              \(21.\)已知\(F_{1}\),\(F_{2}\)是椭圆\( \dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1(a > b > 0)\)的两个焦点,离心率为\( \dfrac{1}{2}\),\(P\)为椭圆上的一点,且\(∠F_{1}PF_{2}=60^{\circ}\),\(\triangle PF_{1}F_{2}\)的面积为\( \sqrt{3}\).


               \((1)\)求椭圆的方程;

              \((2)\)若直线\(l\):\(y=- \dfrac{1}{2}x+m\)与椭圆交于\(A\),\(B\)两点,与以\(F_{1}F_{2}\)为直径的圆交于\(C\),\(D\)两点,且满足\( \dfrac{|AB|}{|CD|}= \dfrac{5 \sqrt{3}}{4}\),求直线\(l\)的方程.

            • 2.

              给出下列命题:

              \(①\)已知圆\(C:x^{2}+y^{2}=1\)外一点\(P(3,4)\),过点\(P\)作圆\(C\)的切线,切点分别为点\(A\)、\(B\),则\(AB\)所在的直线方程为\(3x+4y-2=0\);

              \(②\)已知\(BC\)是圆\(x^{2}+y^{2}=25\)的动弦,且\(|BC|=6\),则\(BC\)的中点的轨迹方程是\(x^{2}+y^{2}=16\);

              \(③\)已知\(A\)、\(B\)两点的坐标分别为\(A(x_{1},y_{1})\)、\(B(x_{2},y_{2})\),则以\(AB\)为直径的圆的方程为:\((x-x_{1})(x-x_{2})+(y-y_{1})(y-y_{2})=0\);

              \(④\)已知直角坐标系中圆\(C\)方程为\(F(x,y)=0\),\(P(x_{0},y_{0})\)为圆内一点\((\)非圆心\()\),那么方程\(F(x,y)=F(x_{0},y_{0})\)所表示的曲线是比圆\(C\)半径小,与圆\(C\)同心的圆;

              \(⑤\)曲线\(x^{2}+y^{2}-|x|-|y|=0\)围成的图形的面积为\(π\).

              其中正确的命题为_________.

            • 3.

              已知圆\(C\)的圆心在\(x\)轴的正半轴上,点\(M(0,\sqrt{5})\)在圆\(C\)上,且圆心到直线\(2x-y=0\)的距离为\(\dfrac{4\sqrt{5}}{5}\),那么圆\(C\)的方程为________.

            • 4.

              方程\(y= \sqrt{1-x^{2}}\)表示的曲线是\((\)  \()\)

              A.上半圆                      
              B.下半圆

              C.圆                                              
              D.抛物线
            • 5.

              已知圆\(O\):\(x^{2}+y^{2}=4\),点\(A(- \sqrt{3},0)\),\(B( \sqrt{3},0)\),以线段\(AP\)为直径的圆\(C_{1}\)内切于圆\(O.\)记点\(P\)的轨迹为\(C_{2}\).

              \((1)\)证明:\(|AP|+|BP|\)为定值,并求\(C_{2}\)的方程;

              \((2)\)过点\(O\)的一条直线交圆\(O\)于\(M\),\(N\)两点,点\(D(-2,0)\),直线\(DM\),\(DN\)与\(C_{2}\)的另一个交点分别为\(S\),\(T.\)记\(\triangle DMN\),\(\triangle DST\)的面积分别为\(S_{1}\),\(S_{2}\),求\( \dfrac{S_{1}}{S_{2}}\)的取值范围.

            • 6.

              若点集\(A=\{(x,y)|{{x}^{2}}+{{y}^{2}}\leqslant 1\},B=\{(x,y)|-1\leqslant x\leqslant 1,-1\leqslant y\leqslant 1\}\),则点集\(P=\left\{ (x,y)\left| x={{x}_{1}}+1,y={{y}_{1}}+1 \right. \right.,({{x}_{1}},{{y}_{1}})\in A\}M=\{(x,y)|x={x}_{1}+{x}_{2},y={y}_{1}+{y}_{2} ,(x_{1},y_{1})∈A,({x}_{2},{y}_{2})∈B\} \)所表示的区域的面积分别为_______________;    _______________\(.\) 

            • 7.

              已知圆\(C\)与直线\(y=x\)及\(x-y-4=0\)都相切,圆心在直线\(y=-x\)上,则圆\(C\)的方程为\((\)    \()\)

              A.\((x+1)^{2}+(y-1)^{2}=2\)
              B.\((x+1)^{2}+(y+1)^{2}=2\)
              C.\((x-1)^{2}+(y-1)^{2}=2\)
              D.\((x-1)^{2}+(y+1)^{2}=2\)
            • 8.

              已知圆\(C\)的圆心在\(x\)轴的正半轴上,点\(M(0,\sqrt{5})\)在圆\(C\)上,且圆心到直线\(2x-y=0\)的距离为\(\dfrac{4\sqrt{5}}{5}\),则圆\(C\)的方程为____.

            • 9.

              如图所示,已知\(A\)、\(B\)、\(C\)是长轴长为\(4\)的椭圆\(E\)上的三点,点\(A\)是长轴的一个端点,\(BC\)过椭圆中心\(O\),且\( \overrightarrow{AC}· \overrightarrow{BC}=0 \),\(|BC|=2|AC|\).


              \((1)\)求椭圆\(E\)的方程;

              \((2)\)在椭圆\(E\)上是否存点\(Q\),使得\(|QB{{|}^{2}}-|QA{{|}^{2}}=2\)?若存在,有几个\((\)不必求出\(Q\)点的坐标\()\),若不存在,请说明理由.

              \((3)\)过椭圆\(E\)上异于其顶点的任一点\(P\),作\(\odot O:{{x}^{2}}+{{y}^{2}}=\dfrac{4}{3}\)的两条切线,切点分别为\(M\)、\(N\),若直线\(MN\)在\(x\)轴、\(y\)轴上的截距分别为\(m\)、\(n\),证明:\(\dfrac{1}{3{{m}^{2}}}+\dfrac{1}{{{n}^{2}}}\)为定值.

            • 10.

              \((1)\)圆\({{x}^{2}}+{{y}^{2}}+2x-4y-3=0\)的圆心坐标为________,半径\(r=\)________;

                  \((2)\)圆\({{x}^{2}}+{{y}^{2}}+2mx=0\)的圆心坐标为________,半径\(r=\)________.

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