
,

,

为常数,离心率为

的双曲线

:

上的动点

到两焦点的距离之和的最小值为

,抛物线

:


的焦点与双曲线

的一顶点重合。(Ⅰ)求抛物线

的方程;(Ⅱ)过直线

:

(

为负常数)上任意一点

向抛物线

引两条切线,切点分别为
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、

,坐标原点

恒在以

为直径的圆内,求实数

的取值范围。
(Ⅰ)

(Ⅱ)

第一问中利用由已知易得双曲线焦距为

,离心率为

,则长轴长为2,故双曲线的上顶点为

,所以抛物线

的方程

第二问中,

为

,

,

,
故直线

的方程为

,即

,
所以

,同理可得:

借助于根与系数的关系得到即

,

是方程

的两个不同的根,所以

由已知易得

,即

解:(Ⅰ)由已知易得双曲线焦距为

,离心率为

,则长轴长为2,故双曲线的上顶点为

,所以抛物线

的方程

(Ⅱ)设

为

,

,

,
故直线

的方程为

,即

,
所以

,同理可得:

,
即

,

是方程

的两个不同的根,所以

由已知易得

,即

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设

是双曲线

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
作与

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
,满足

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
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
表示双曲线,则

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垂直于

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
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
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
为圆心的圆与
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,圆
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与

轴相交于

两点.若

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