已知橢圓

的離心率為

,長軸長為

,直線

交橢圓于不同的兩點A、B.
(1)求橢圓的方程;
(2)求

的值(O點為坐標原點);
(3)若坐標原點O到直線

的距離為

,求

面積的最大值.
(1)

(2)
(3)

當|AB最大時,

的面積最大值
(1)依題意得

,所以

.橢圓方程為

(2)直線方程與橢圓方程聯(lián)立,保證

,求出

,利用

,可得

(3)由原點O到直線

的距離為

得

.直線方程與橢圓方程聯(lián)立,保證

,求出

,利用

,可得


利用不等式求出最值.注意

的討論.
解:(1)設(shè)橢圓的半焦距為c,依題意

解得

由

2分

所求橢圓方程為

3分
(2)

設(shè)

,其坐標滿足方程

消去

并整理得

4分
則有

,

6分



8分
(3)由已知

,可得

9分
將


代入橢圓方程,
整理得



10分


11分

12分
當且僅當

,即

時等號成立,經(jīng)檢驗,

滿足(*)式
當

時,

綜上可知

13分

當|AB最大時,

的面積最大值

14分
練習(xí)冊系列答案
相關(guān)習(xí)題
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
如圖,在平面直角坐標系xOy中, 點A為橢圓E:

(

)的左頂點, B,C在橢圓E上,若四邊形OABC為平行四邊形,且∠OAB=30°,則橢圓E的離心率等于 .

查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
動點
A到定點

和

的距離的和為4,則動點
A的軌跡為 ( )
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖,橢圓

的離心率為

,直線

和

所圍成的矩形ABCD的面積為8.

(Ⅰ)求橢圓M的標準方程;
(Ⅱ) 設(shè)直線

與橢圓M有兩個不同的交點

與矩形ABCD有兩個不同的交點

.求

的最大值及取得最大值時m的值.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知橢圓

的離心率

,點F為橢圓的右焦點,點A、B分別為橢圓的左、右頂點,點M為橢圓的上頂點,且滿足

(1)求橢圓C的方程;
(2)是否存在直線

,當直線

交橢圓于P、Q兩點時,使點F恰為

的垂心(三角形三條高的交點)?若存在,求出直線

方程;若不存在,請說明理由。

查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知圓方程為

(1)求圓心軌跡的參數(shù)方程

和普通方程;
(2)點

是(1)中曲線

上的動點,求

的取值范圍.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
設(shè)

是橢圓

的兩個焦點,

是橢圓上的動點(不能重合于長軸的兩端點),

是

的內(nèi)心,直線

交

軸于點

,則
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
若橢圓

的左右焦點分別為

,線段

被拋物線

的焦點

內(nèi)分成了

的兩段.
(1)求橢圓的離心率;
(2)過點

的直線

交橢圓于不同兩點

、

,且

,當

的面積最大時,求直線

的方程.

查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
橢圓

的兩焦點之間的距離為 ( )
查看答案和解析>>