設(shè)橢圓C:
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過點(0,4),離心率為
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(Ⅰ)求C的方程;(Ⅱ)求過點(3,0)且斜率為

的直線被C所截線段的長度.
試題分析:(1)橢圓的方程是標(biāo)準(zhǔn)方程,已知橢圓過點
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,這必定是橢圓的頂點,從而易知
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(當(dāng)然也可直接把
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代入橢圓方程解出
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),再由離心率為
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,可求出

.得橢圓的方程.(2)這是直線與橢圓相交求相交弦長的問題,我們可以用相交弦長公式
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求解,這里
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是直線的斜率,

是交點的橫坐標(biāo).
試題解析:(Ⅰ)將(0,4)代入C的方程得
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∴
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,又

得

即

,
∴

∴C的方程為

.
( Ⅱ)過點

且斜率為

的直線方程為

,
設(shè)直線與C的交點為A

,B
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,將直線方程

代入C的方程,得

,即

,
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,
∴
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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
(本小題滿分12分)已知

的兩頂點坐標(biāo)

,
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,圓
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是

的內(nèi)切圓,在邊

,
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,
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上的切點分別為
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,
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(從圓外一點到圓的兩條切線段長相等),動點

的軌跡為曲線

.

(1)求曲線

的方程;
(2)設(shè)直線

與曲線

的另一交點為

,當(dāng)點

在以線段

為直徑的圓上時,求直線

的方程.
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知點
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(
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,

是常數(shù)),且動點

到

軸的距離比到點

的距離小

.
(1)求動點

的軌跡

的方程;
(2)(i)已知點

,若曲線

上存在不同兩點

、

滿足

,求實數(shù)

的取值范圍;
(ii)當(dāng)

時,拋物線

上是否存在異于

、

的點

,使得經(jīng)過
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、

、

三點的圓和拋物線

在點

處有相同的切線,若存在,求出點

的坐標(biāo),若不存在,請說明理由.
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科目:高中數(shù)學(xué)
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已知橢圓中心在原點,焦點在

軸上,焦距為2,離心率為

(1)求橢圓的方程;
(2)設(shè)直線

經(jīng)過點

(0,1),且與橢圓交于

兩點,若

,求直線

的方程.
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如圖,在平面直角坐標(biāo)系

中,已知橢圓

經(jīng)過點


,橢圓的離心率

.

(1)求橢圓

的方程;
(2)過點

作兩直線與橢圓

分別交于相異兩點

、

.若

的平分線與

軸平行, 試探究直線

的斜率是否為定值?若是, 請給予證明;若不是, 請說明理由.
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曲線

在矩陣

的變換作用下得到曲線

.
(Ⅰ)求矩陣

;
(Ⅱ)求矩陣

的特征值及對應(yīng)的一個特征向量.
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題型:解答題
已知橢圓

的離心率為

,橢圓的短軸端點與雙曲線

的焦點重合,過點

且不垂直于
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軸直線
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與橢圓

相交于
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、

兩點.
(Ⅰ)求橢圓

的方程;
(Ⅱ)求

的取值范圍.
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科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
已知動點

到點

的距離等于它到直線

的距離,則點

的軌跡方程是
.
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科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
已知橢圓

的離心率為
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,過右焦點
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且斜率為
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的直線與
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相交于
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兩點.若
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,則
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( )
A.1 | B. | C. | D.2 |
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