拋物線

與直線

相切,

是拋物線上兩個動點,

為拋物線的焦點,

的垂直平分線

與

軸交于點

,且

.
(1)求

的值;
(2)求點

的坐標;
(3)求直線

的斜率

的取值范圍.
試題分析:(1)將拋物線

與直線

聯(lián)立,消元后得到

有兩個相等實根,由

求得

.
(2)利用,拋物線

的準線

且

,結(jié)合定義可得

.
由

在

的垂直平分線上,得到

,可以建立

橫坐標的方程,通過解方程得到解題目的.
(3)點

在拋物線

的內(nèi)部,應有

,設直線

方程

后,據(jù)此可建立

的不等式,進一步確定

的取值范圍為

.
試題解析:
(1)由

得:

有兩個相等實根 1分
即

得:

為所求 3分
(2)拋物線

的準線

且

,
由定義得

,則

5分
設

,由

在

的垂直平分線上,從而

6分
則



8分
因為

,所以

又因為

,所以

,則點

的坐標為

10分
(3)設

的中點

,有

11分
設直線

方程

過點

,得

12分
又因為點

在拋物線

的內(nèi)部,則

13分
得:

,則

又因為

,則

故

的取值范圍為

14分
練習冊系列答案
相關(guān)習題
科目:高中數(shù)學
來源:不詳
題型:解答題
矩形

的中心在坐標原點,邊

與

軸平行,

=8,

=6.

分別是矩形四條邊的中點,

是線段

的四等分點,

是線段

的四等分點.設直線

與

,

與

,

與

的交點依次為

.

(1)求以

為長軸,以

為短軸的橢圓Q的方程;
(2)根據(jù)條件可判定點

都在(1)中的橢圓Q上,請以點L為例,給出證明(即證明點L在橢圓Q上).
(3)設線段

的

(

等分點從左向右依次為

,線段

的

等分點從上向下依次為

,那么直線

與哪條直線的交點一定在橢圓Q上?(寫出結(jié)果即可,此問不要求證明)
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
在平面直角坐標系

中,已知橢圓

:

的離心率

,且橢圓C上一點

到點Q

的距離最大值為4,過點

的直線交橢圓

于點

(Ⅰ)求橢圓C的方程;
(Ⅱ)設P為橢圓上一點,且滿足

(O為坐標原點),當

時,求實數(shù)

的取值范圍.
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
已知

是橢圓

的右焦點,圓

與

軸交于

兩點,

是橢圓

與圓

的一個交點,且
(Ⅰ)求橢圓

的離心率;
(Ⅱ)過點

與圓

相切的直線

與

的另一交點為

,且

的面積為

,求橢圓

的方程
查看答案和解析>>
科目:高中數(shù)學
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) 設直線l過橢圓的右焦點F
2(l不垂直坐標軸),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
來源:不詳
題型:解答題
已知

為拋物線

的焦點,拋物線上點

滿足


(Ⅰ)求拋物線

的方程;
(Ⅱ)

點的坐標為(

,

),過點F作斜率為

的直線與拋物線交于

、

兩點,

、

兩點的橫坐標均不為

,連結(jié)

、

并延長交拋物線于

、

兩點,設直線

的斜率為

,問

是否為定值,若是求出該定值,若不是說明理由.
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:填空題
在平面直角坐標系中,動點

到兩條坐標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
來源:不詳
題型:填空題
已知拋物線

的準線過雙曲線

的一個焦點, 且雙曲線的離心率為2, 則該雙曲線的方程為
.
查看答案和解析>>
科目:高中數(shù)學
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>>