已知椭圆

:

的一个焦点为

,离心率为

.设

是椭圆

长轴上的一个动点,过点

且斜率为

的直线

交椭圆于

,

两点.
(1)求椭圆

的方程;
(2)求

的最大值.
(1)

;(2)

.
试题分析:(1)由题意,

,

,根据

求出

,则椭圆的方程为

. (2)设点

(

),则直线

的方程为

,联立

得

,而


,带入韦达定理

,

,则


,而

, 即

,则当

时,

,

的最大值为

.
试题解析:(1)由已知,

,

,
∴

,

3分
∴ 椭圆的方程为

. 4分
(2)设点

(

),则直线

的方程为

, 2分
由

消去

,得

4分
设

,

,则

,

6分
∴




8分
∵

, 即

∴当

时,

,

的最大值为

. 10分
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如图,已知点

为椭圆


右焦点,圆


与椭圆

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
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
与圆

相切于点

.

(1)求

的值及椭圆

的标准方程;
(2)设动点

满足

,其中M、N是椭圆

上的点,

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
,求证:

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
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
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
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
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
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为常数,则动点

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已知直线

与椭圆

相交于

、

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
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
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
的右焦点为

,椭圆

与

轴正半轴交于
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点,与
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轴正半轴交于
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,且
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,则椭圆
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的方程为( )
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