在平面直角坐標(biāo)系

中,以坐標(biāo)原點

為極點,

軸的非負半軸為極軸建立極坐標(biāo)系.已知曲線

的極坐標(biāo)方程為

,直線

的參數(shù)方程為

為參數(shù),

).
(Ⅰ)化曲線

的極坐標(biāo)方程為直角坐標(biāo)方程;
(Ⅱ)若直線

經(jīng)過點

,求直線

被曲線

截得的線段

的長.
(Ⅰ)

;(Ⅱ)

.
試題分析:(Ⅰ)先在方程

兩邊同時乘以

得

,然后將

,

進行代換,邊可以得到曲線的直角坐標(biāo)方程;(Ⅱ)將直線的方程與拋物線方程進行聯(lián)立,然后利用焦點弦公式

并結(jié)合韋達定理可以求出

試題解析:解法一:(Ⅰ)由

得,

,
即曲線

的直角坐標(biāo)方程為

. 3分
(Ⅱ)由直線

經(jīng)過點

,得直線

的直角坐標(biāo)方程是

,
聯(lián)立

,消去

,得

,又點

是拋物線的焦點,
由拋物線定義,得弦長

. 7分
解法二:(Ⅰ)同解法一. 3分
(Ⅱ)由直線

經(jīng)過點

,得

,直線

的參數(shù)方程為

將直線

的參數(shù)方程代入

,得

,
所以

. 7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:解答題
設(shè)橢圓

:

的左、右焦點分別是

、

,下頂點為

,線段

的中點為

(

為坐標(biāo)原點),如圖.若拋物線

:

與

軸的交點為

,且經(jīng)過

、

兩點.

(Ⅰ)求橢圓

的方程;
(Ⅱ)設(shè)

,

為拋物線

上的一動點,過點

作拋物線

的切線交橢圓

于

、

兩點,求

面積的最大值.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
知橢圓

的左右焦點為F
1,F(xiàn)
2,離心率為

,以線段F
1 F
2為直徑的圓的面積為

, (1)求橢圓的方程;(2) 設(shè)直線l過橢圓的右焦點F
2(l不垂直坐標(biāo)軸),且與橢圓交于A、B兩點,線段AB的垂直平分線交x軸于點M(m,0),試求m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:解答題
已知橢圓


(a>b>0)拋物線


,從每條曲線上取兩個點,將其坐標(biāo)記錄于下表中:

(1)求

的標(biāo)準(zhǔn)方程;
(2)四邊形ABCD的頂點在橢圓

上,且對角線AC、BD過原點O,若

,
(i) 求

的最值.
(ii) 求四邊形ABCD的面積;
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
在平面直角坐標(biāo)系中,動點

到兩條坐標(biāo)軸的距離之和等于它到點

的距離,記點

的軌跡為曲線

.
(I) 給出下列三個結(jié)論:
①曲線

關(guān)于原點對稱;
②曲線

關(guān)于直線

對稱;
③曲線

與

軸非負半軸,

軸非負半軸圍成的封閉圖形的面積小于

;
其中,所有正確結(jié)論的序號是_____;
(Ⅱ)曲線

上的點到原點距離的最小值為______.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
設(shè)

,則方程

不能表示的曲線為( )
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
拋物線

的焦點為

,

在拋物線上,且

,弦

的中點

在其準(zhǔn)線上的射影為

,則

的最大值為

查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
雙曲線

的左、右焦點分別為

和

,左、右頂點分別為

和

,過焦點

與

軸垂直的直線和雙曲線的一個交點為

,若

是

和

的等比中項,則該雙曲線的離心率為
.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
已知

,

分別是雙曲線

:


的兩個焦點,雙曲線

和圓

:

的一個交點為

,且

,那么雙曲線

的離心率為 ( )
查看答案和解析>>