已知椭圆C
1的方程为
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678021_ST/0.png)
+y
2=1,双曲线C
2的左、右焦点分别是C
1的左、右顶点,而C
2的左、右顶点分别是C
1的左、右焦点.
(1)求双曲线C
2的方程;
(2)若直线l:y=kx+
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678021_ST/1.png)
与双曲线C
2恒有两个不同的交点A和B,且
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678021_ST/2.png)
•
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678021_ST/3.png)
>2(其中O为原点),求k的取值范围.
考点分析:
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如图,直角三角形ABC的顶点坐标A(-2,0),直角顶点
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678020_ST/0.png)
,顶点C在x轴上,点P为线段OA的中点.
(1)求BC边所在直线方程;
(2)M为直角三角形ABC外接圆的圆心,求圆M的方程;
(3)若动圆N过点P且与圆M内切,求动圆N的圆心N的轨迹方程.
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如图,
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为半圆,AB为半圆直径,O为半圆圆心,且OD⊥AB,Q为线段OD的中点,已知|AB|=4,曲线C过Q点,动点P在曲线C上运动且保持|PA|+|PB|的值不变.
(Ⅰ)建立适当的平面直角坐标系,求曲线C的方程;
(Ⅱ)过点B的直线l与曲线C交于M、N两点,与OD所在直线交于E点,若
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678019_ST/1.png)
为定值.
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已知椭圆
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678018_ST/0.png)
=1(a>b>0)的离心率e=
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678018_ST/1.png)
,左、右焦点分别为F
1、F
2,点
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678018_ST/2.png)
,点F
2在线段PF
1的中垂线上.
(1)求椭圆C的方程;
(2)设直线l:y=kx+m与椭圆C交于M、N两点,直线F
2M与F
2N的倾斜角分别为α,β,且α+β=π,求证:直线l过定点,并求该定点的坐标.
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给定椭圆
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678017_ST/0.png)
,称圆心在原点O,半径为
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678017_ST/1.png)
的圆是椭圆C的“准圆”.若椭圆C的一个焦点为
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678017_ST/2.png)
,其短轴上的一个端点到F的距离为
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678017_ST/3.png)
.
(I)求椭圆C的方程和其“准圆”方程.(II)点P是椭圆C的“准圆”上的一个动点,过点P作直线l
1,l
2,使得l
1,l
2与椭圆C都只有一个交点,且l
1,l
2分别交其“准圆”于点M,N.
①当P为“准圆”与y轴正半轴的交点时,求l
1,l
2的方程;
②求证:|MN|为定值.
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已知向量
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678016_ST/0.png)
=(0,x),
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678016_ST/1.png)
=(1,1),
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678016_ST/2.png)
=(x,0),
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678016_ST/3.png)
=(y
2,1)(其中x,y是实数),又设向量
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678016_ST/4.png)
,
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678016_ST/5.png)
,且
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678016_ST/6.png)
,点P(x,y)的轨迹为曲线C.
(1)求曲线C的方程;
(2)设曲线C与y轴的正半轴的交点为M,过点M作一条直线l与曲线C交于另一点N,当|MN|=
![manfen5.com 满分网](http://img.manfen5.com/res/GZSX/web/STSource/20131025123721212967866/SYS201310251237212129678016_ST/7.png)
时,求直线 l 的方程.
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