(本題15分)已知曲線
C是到點
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和到直線
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距離相等的點的軌跡,
l是過點
Q(-1,0)的直線,
M是
C上(不在
l上)的動點;
A、B在
l上,
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軸(如圖)。
(Ⅰ)求曲線
C的方程;
(Ⅱ)求出直線
l的方程,使得
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為常數(shù)。
(Ⅰ)
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,(Ⅱ)
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本題主要考查求曲線軌跡方程,兩條直線的位置關(guān)系等基礎(chǔ)知識,考查解析幾何的基本思想方法和綜合解題能力。滿分15分。
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(I)設(shè)
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為
C上的點,則
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.
N到直線

的距離為
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.
由題設(shè)得
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.
化簡,得曲線
C的方程為
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.
(II)解法一:
設(shè)
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,直線
l:

,則
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,從而

.
在Rt△
QMA中,因為
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,
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.
所以
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
,
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當
k=2時,
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從而所求直線
l方程為
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解法二:
設(shè)
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,直線直線
l:

,則
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,從而
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過
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垂直于
l的直線
l1:

,
因為
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,所以
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,
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,
當
k=2時,

,
從而所求直線
l方程為
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
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A(
x
,
y
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x
,
y
) 是橢圓

(
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
,

= (
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,

),且滿足
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·

= 0,橢圓的離心率
e =
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


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上

的一個動點,弦AB、AC分別過焦點

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
∶

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
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
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
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
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
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
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
作

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
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
的切線經(jīng)過橢圓的右焦點

.
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
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
,使得
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
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
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
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若拋物線

的焦點與雙曲線

的右焦點重合,則
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的值為C

.

.

.

.
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