已知橢圓

的中心在原點

,離心率
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,右焦點為

.
(Ⅰ)求橢圓

的方程;
(Ⅱ)設橢圓的上頂點為

,在橢圓
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上是否存在點

,使得向量
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與
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共線?若存在,求直線
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的方程;若不存在,簡要說明理由.
試題分析:(Ⅰ) 由離心率和焦點坐標兩個條件求出橢圓的C的方程.
(Ⅱ)首先假設存在點P,再通過向量
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與
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共線.得到關于一個關于點P

的橫縱坐標的
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的一個等式.因為點P
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在橢圓上,所以又得到一個關于
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的一個方程.由此可解出
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的值.從而寫出直線AP的方程.本小題是橢圓中的一個較簡單的問題,通過兩個已知條件求出橢圓的方程.接著利用橢圓方程以及向量的共線知識,求出共線問題.
試題解析:(1)設橢圓
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的方程為

,
離心率

,右焦點為
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,
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
,
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
,
故橢圓

的方程為

6分
(2)假設橢圓

上存在點
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(

),使得向量
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與
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共線,
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
,
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, 7分
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
(1) 8分
又
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點
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(
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)在橢圓
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上,
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
(2) 9分
由(1)、(2)組成方程組解得:
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,或

, 10分
當點

的坐標為
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時,直線

的方程為
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, 11分
當點

的坐標為
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時,直線

的方程為

, 12分
故直線

的方程為
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或
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13分
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已知橢圓


上的點到其兩焦點距離之和為

,且過點

.
(Ⅰ)求橢圓方程;
(Ⅱ)

為坐標原點,斜率為

的直線過橢圓的右焦點,且與橢圓交于點

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
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
,求△

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
的兩個焦點為F
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2,橢圓上一點M

滿足

.
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
與橢圓恒有不同交點A,B,且

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定義:對于兩個雙曲線

,

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的實軸是

的虛軸,

的虛軸是

的實軸,則稱

,

為共軛雙曲線.現(xiàn)給出雙曲線

和雙曲線

,其離心率分別為

.
(1)寫出

的漸近線方程(不用證明);
(2)試判斷雙曲線

和雙曲線

是否為共軛雙曲線?請加以證明.
(3)求值:

.
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已知拋物線
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的焦點為

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
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為拋物線C上的一點,且

的外接圓圓心到準線的距離為

.

(I)求拋物線C的方程;
(II)若圓F的方程為

,過點P作圓F的2條切線分別交

軸于點

,求

面積的最小值時

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已知橢圓的中心在原點,焦點在

軸上,離心率為

,長軸長為

,直線

交橢圓于不同的兩點

.
(1)求橢圓的方程;
(2)求

的取值范圍;
(3)若直線
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不經(jīng)過橢圓上的點
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,求證:直線
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的斜率互為相反數(shù).
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科目:高中數(shù)學
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已知橢圓
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:
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的左、右焦點和短軸的兩個端點構成邊長為2的正方形.
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(Ⅰ)求橢圓
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的方程;
(Ⅱ)過點
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的直線
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與橢圓
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相交于
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,
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兩點.點
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,記直線
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的斜率分別為
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,當
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最大時,求直線
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的方程.
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科目:高中數(shù)學
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設AB是橢圓的長軸,點C在橢圓上,且
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,若AB=4,
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,則橢圓的兩個焦點之間的距離為________.
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科目:高中數(shù)學
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以拋物線
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的焦點為圓心,且與雙曲線
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的兩條漸近線都相切的圓的方程為
.
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