.(本小題滿分12分)
已知橢圓

的離心率為

,且經(jīng)過(guò)點(diǎn)

(1)求橢圓C的方程;
(2)已知A為橢圓C的左頂點(diǎn),直線

過(guò)右焦點(diǎn)F與橢圓C交于M,N兩點(diǎn),若AM、AN的斜率

滿足

(定值

),求直線

的斜率。
解:(1)


2分
又

解得


3分
橢圓C的方程是

…………………………4分
(2)若直線

斜率不存在,顯然

不合題意 ………………………………5分
設(shè)直線方程為

取立方程組

得

……………………………………7分

……………………………………8分
又





……………………………………11分


………………………………………12分
練習(xí)冊(cè)系列答案
相關(guān)習(xí)題
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本小題滿分14分)已知橢圓

的中心在坐標(biāo)原點(diǎn),焦點(diǎn)在

軸上,長(zhǎng)軸長(zhǎng)為

,離心率為

,經(jīng)過(guò)其左焦點(diǎn)

的直線

交橢圓

于

、

兩點(diǎn)(I)求橢圓

的方程;
(II)在

軸上是否存在一點(diǎn)

,使得

恒為常數(shù)?若存在,求出

點(diǎn)的坐標(biāo)和這個(gè)常數(shù);若不存在,說(shuō)明理由.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:單選題
已知雙曲線

的左、右焦點(diǎn)分別為

、

,點(diǎn)

在雙曲線的右支上,直線

為過(guò)

且切于雙曲線的直線,且平分

,過(guò)

作與直線

平行的直線交

于

點(diǎn),則

,利用類比推理:若橢圓

的左、右焦點(diǎn)分別為

、

,點(diǎn)

在橢圓上,直線

為過(guò)

且切于橢圓的直線,且平分

的外角,過(guò)

作與直線平行的直線交

于

點(diǎn),則

的值為 ( )
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本題滿分12分)
如圖6,在平面直角坐標(biāo)系

中,設(shè)點(diǎn)

,直線

:

,點(diǎn)

在直線

上移動(dòng),

是線段

與

軸的交點(diǎn),

.

(I)求動(dòng)點(diǎn)

的軌跡的方程

;
(II)設(shè)圓

過(guò)

,且圓心

在曲線

上,

是圓

在

軸上截得的弦,當(dāng)

運(yùn)動(dòng)時(shí)弦長(zhǎng)

是否為定值?請(qǐng)說(shuō)明理由.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
已知中心在原點(diǎn)的雙曲線C的右焦點(diǎn)為(2,0),實(shí)軸長(zhǎng)為2
(1)求雙曲線C的方程;
(2)若直線l:y=kx+與雙曲線C左支交于A、B兩點(diǎn),求k的取值范圍
(3)在(2)的條件下,線段AB的垂直平分線l0與y軸交于M(0,m),求m的取值范圍
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本小題滿分12分)
已知定點(diǎn)A(12,0),M為曲線

上的動(dòng)點(diǎn),(1)若

,試求動(dòng)點(diǎn)P的
軌跡C的方程.2)若

與曲線C相交于不同的兩點(diǎn)E、F, O為坐標(biāo)原點(diǎn)且

,求∠EOF的余弦值和實(shí)數(shù)

的值.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:單選題
若拋物線

的焦點(diǎn)與雙曲線

的右焦點(diǎn)重合,則

的值為( )
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
如圖邊長(zhǎng)為2的正方形花園的一角是以A為中心,1為半徑的扇形水池.現(xiàn)需在其余部分設(shè)計(jì)一個(gè)矩形草坪PNCQ,其中P是水池邊上任意一點(diǎn),點(diǎn)N、Q分別在邊BC和CD上,設(shè)∠PAB為θ.
(I)用θ表示矩形草坪PNCQ的面積,并求其最小值;
(II)求點(diǎn)P到邊BC和AB距離之比

的最小值.

查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
本小題滿分12分)
如圖,已知橢圓C1的中心在原點(diǎn)O,長(zhǎng)軸左、右端點(diǎn)M,N在x軸上,橢圓C2的短軸為MN,且C1,C2的離心率都為e,直線l⊥MN,l與C1交于兩點(diǎn),與C2交于兩點(diǎn),這四點(diǎn)按縱坐標(biāo)從大到小依次為A,B,C,D.

(1)設(shè)

,求

與

的比值;
(2)當(dāng)e變化時(shí),是否存在直線l,使得BO∥AN,并說(shuō)明理由
查看答案和解析>>