在直角坐標(biāo)系

中,點
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到點
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,
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的距離之和是

,點

的軌跡
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與
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軸的負半軸交于點
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,不過點

的直線

與軌跡

交于不同的兩點

和

.
⑴求軌跡

的方程;
⑵當(dāng)

時,證明直線

過定點.
⑴∵點
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到
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,
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的距離之和是
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,∴
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的軌跡
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是長軸為
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,焦點在
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軸上焦距為
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的橢圓,其方程為

.
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⑵將
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,代入曲線
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的方程,整理得
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,因為直線
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與曲線
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交于不同的兩點
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和
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,所以
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①
設(shè)
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,
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,則
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,
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②
且
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③
顯然,曲線
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與
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軸的負半軸交于點
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,所以
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,
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.由
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,得
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.
將②,③代入上式,整理得

.所以

,即

或

.經(jīng)檢驗,都符合條件①,
當(dāng)

時,直線

的方程為

.顯然,此時直線

經(jīng)過定點
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點.即直線

經(jīng)過點

,與題意不符.
當(dāng)
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時,直線

的方程為
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.顯然,此時直線

經(jīng)過定點
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點,且不過點

.
綜上,
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與
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的關(guān)系是:

,且直線

經(jīng)過定點
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點.
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直線
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分拋物線
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與
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軸所圍成圖形為面積相等的兩個部分,求

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直線y=x-
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被橢圓x
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(本題15分)已知曲線

與曲線

,設(shè)點

是曲線

上任意一點,直線

與曲線

交于

、

兩點.
(1)判斷直線

與曲線

的位置關(guān)系;
(2)以

、

兩點為切點分別作曲線

的切線,設(shè)兩切線的交點為

,求證:點

到直線

:

與

:

距離的乘積為定值.
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已知拋物線

:
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,焦點為
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,其準(zhǔn)線與

軸交于點

;橢圓

:分別以

為左、右焦點,其離心率

;且拋物線

和橢圓


的一個交點記為

.
(1)當(dāng)

時,求橢圓

的標(biāo)準(zhǔn)方程;

(2)在(1)的條件下,若直線

經(jīng)過橢圓

的右焦點
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,且
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與拋物線
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相交于
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兩點,若弦長
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等于
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的周長,求直線
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的方程
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.
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
在圓
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上任取一點
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,過點
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作
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軸的垂線段
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,
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為垂足,當(dāng)點
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在圓上運動時,線段
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的中點
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的軌跡為曲線
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(Ⅰ)求曲線
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的方程;
(Ⅱ)過點
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
的直線
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與曲線
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相交于不同的兩點
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, 點
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在線段
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的垂直平分線上,且
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,求
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的值
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科目:高中數(shù)學(xué)
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已知橢圓
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,拋物線
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,點
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是
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上的動點,過點
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作拋物線
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的切線
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,交橢圓
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于
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兩點,
(1)當(dāng)
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的斜率是
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時,求
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;
(2)設(shè)拋物線
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的切線方程為
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,當(dāng)
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是銳角時,求
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的取值范圍.
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科目:高中數(shù)學(xué)
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題型:填空題
曲線C:
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與
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軸的交點關(guān)于原點的對稱點稱為“望點”,以“望點”為圓心,凡是與曲線C有公共點的圓,皆稱之為“望圓”,則當(dāng)a=1,b=1時,所有的“望圓”中,面積最小的“望圓”的面積為
.
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科目:高中數(shù)學(xué)
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題型:填空題
橢圓
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的焦點在
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軸上,長軸長是短軸長的兩倍,則
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的值為________
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