已知橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454424313.png)
的中心在原點(diǎn),焦點(diǎn)在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454440266.png)
軸上,一條經(jīng)過點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454455532.png)
且方向向量為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454471648.png)
的直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454486280.png)
交橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454424313.png)
于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454518423.png)
兩點(diǎn),交
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454440266.png)
軸于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454564399.png)
點(diǎn),且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454580693.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240054545964467.jpg)
(1)求直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454486280.png)
的方程;
(2)求橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454424313.png)
長(zhǎng)軸長(zhǎng)的取值范圍.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454658743.png)
;(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454689692.png)
.
試題分析:(1)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454486280.png)
過點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454455532.png)
且方向向量為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454471648.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454767584.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454486280.png)
方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454798958.png)
,
化簡(jiǎn)為:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454830797.png)
∴直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454486280.png)
的方程為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454658743.png)
(2)設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454830797.png)
和橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240054549081085.png)
交于兩點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455064858.png)
,和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454440266.png)
軸交于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455095606.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455110695.png)
,知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455142527.png)
,
將
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455157689.png)
代入
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455173745.png)
中,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240054552041385.png)
……①
由韋達(dá)定理知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240054552203021.png)
由②
2/③知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455235989.png)
,化為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455251938.png)
……④
∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240054552821641.png)
,
化簡(jiǎn),得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455298982.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455298700.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240054553131037.png)
,注意到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455329498.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455344655.png)
又橢圓的焦點(diǎn)在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454440266.png)
軸上,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455376482.png)
,
由④知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240054553911077.png)
,結(jié)合
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455407457.png)
,求得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455422648.png)
.
因此所求橢圓長(zhǎng)軸長(zhǎng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455438362.png)
范圍為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005454689692.png)
.
點(diǎn)評(píng):中檔題,涉及橢圓與直線位置關(guān)系問題,往往利用韋達(dá)定理。本題借助于韋達(dá)定理,建立方程組后,整理得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005455251938.png)
,進(jìn)一步利用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240054553131037.png)
求得a的范圍。
練習(xí)冊(cè)系列答案
相關(guān)習(xí)題
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知直線l經(jīng)過點(diǎn)(0,-2),其傾斜角是60°.
(1)求直線l的方程;
(2)求直線l與兩坐標(biāo)軸圍成三角形的面積.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
方程
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005806117847.png)
表示焦點(diǎn)在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005806133310.png)
軸的雙曲線,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005806148312.png)
的取值范圍是( )
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
以橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005701174509.png)
內(nèi)的點(diǎn)M(1,1)為中點(diǎn)的弦所在直線的方程為( )
A.4x-y-3=0 | B.x-4y+3=0 |
C.4x+y-5=0 | D.x+4y-5=0 |
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
以雙曲線:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005549055654.png)
的右焦點(diǎn)為圓心,并與其漸近線相切的圓的標(biāo)準(zhǔn)方程是______
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
已知雙曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240054320691110.png)
,若過右焦點(diǎn)F且傾斜角為30°的直線與雙曲線的右支有兩個(gè)交點(diǎn),則此雙曲線離心率的取值范圍是__________.
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,在平面直角坐標(biāo)系
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348030413.png)
中,點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348046300.png)
為橢圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348061767.png)
的右頂點(diǎn), 點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348077503.png)
,點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348093395.png)
在橢圓上,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348108554.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348124345.jpg)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240053481394055.jpg)
(1)求直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348155374.png)
的方程;
(2)求直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348155374.png)
被過
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348186493.png)
三點(diǎn)的圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005348202313.png)
截得的弦長(zhǎng);
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
(本小題共14分)
已知橢圓C:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240052270191089.png)
,左焦點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005227035605.png)
,且離心率
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005227051547.png)
(Ⅰ)求橢圓C的方程;
(Ⅱ)若直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005227066888.png)
與橢圓C交于不同的兩點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005227082550.png)
(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005227082550.png)
不是左、右頂點(diǎn)),且以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005227113513.png)
為直徑的圓經(jīng)過橢圓C的右頂點(diǎn)A. 求證:直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824005227144280.png)
過定點(diǎn),并求出定點(diǎn)的坐標(biāo).
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
(本小題滿分12分)已知橢圓的中心在坐標(biāo)原點(diǎn)O,長(zhǎng)軸長(zhǎng)為2
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824004748340344.png)
,離心率e=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824004748372413.png)
,過右焦點(diǎn)F的直線
l交橢圓于P、Q兩點(diǎn).
(Ⅰ)求橢圓的方程;
(Ⅱ)若OP、OQ為鄰邊的平行四邊形是矩形,求滿足該條件的直線
l的方程.
查看答案和解析>>