如圖示:已知拋物線

的焦點為

,過點

作直線

交拋物線

于

、

兩點,經(jīng)過

、

兩點分別作拋物線

的切線

、

,切線

與

相交于點

.

(1)當點

在第二象限,且到準線距離為

時,求

;
(2)證明:

.
(1)

;(2)詳見解析.
試題分析:(1)先利用拋物線的定義求出點

的坐標,然后利用直線

過點

和點

求出直線

的方程,然后將直線

和拋物線的方程聯(lián)立,利用韋達定理與拋物線的定義求出弦

的長;(2)先求出曲線

在點

和點

的切線方程,并求出兩切線的交點

的坐標,驗證

進而得到

.
試題解析:(1)拋物線

的方程為

,則其焦點坐標為

,
設(shè)點

,

,則有

,
由于點

在第二象限,則

,將

代入

得,

,解得

,
故點

的坐標為

,故直線

的方程為

,變形得

,
代入拋物線的方程并化簡得

,由韋達定理得

,

;
(2)設(shè)直線

的方程為

,將

代入拋物線的方程并化簡得

,

對任意

恒成立,
由韋達定理得

,

,
將拋物線的方程化為函數(shù)解析式得,

,則

,
故曲線

在點

處的切線方程為

,即

,即

①,
同理可知,曲線

在點

處的切線方程為

②,
聯(lián)立①②得,

,故點

的坐標為

,

,
而

,

,

.
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
已知拋物線的頂點在坐標原點,焦點為

,點

是點

關(guān)于

軸的對稱點,過點

的直線交拋物線于

兩點。
(Ⅰ)試問在

軸上是否存在不同于點

的一點

,使得

與

軸所在的直線所成的銳角相等,若存在,求出定點

的坐標,若不存在說明理由。
(Ⅱ)若

的面積為

,求向量

的夾角;
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
曲線

在矩陣

的變換作用下得到曲線

.
(Ⅰ)求矩陣

;
(Ⅱ)求矩陣

的特征值及對應(yīng)的一個特征向量.
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
已知圓

直線

與圓

相切,且交橢圓

于

兩點,

是橢圓的半焦距,

,
(Ⅰ)求

的值;
(Ⅱ)O為坐標原點,若

求橢圓

的方程;
(Ⅲ) 在(Ⅱ)的條件下,設(shè)橢圓

的左右頂點分別為A,B,動點

,直線AS,BS與直線

分別交于M,N兩點,求線段MN的長度的最小值.
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
已知橢圓

拋物線

的焦點均在

軸上,

的中心和

的頂點均為坐標原點

從每條曲線上取兩個點,將其坐標記錄于下表中:
(Ⅰ)求分別適合

的方程的點的坐標;
(Ⅱ)求

的標準方程.
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
設(shè)雙曲線

以橢圓

的兩個焦點為焦點,且雙曲線

的一條漸近線是

,
(1)求雙曲線

的方程;
(2)若直線

與雙曲線

交于不同兩點

,且

都在以

為圓心的圓上,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
如圖,曲線

與曲線

相交于

、

、

、

四個點.
⑴ 求

的取值范圍;
⑵ 求四邊形

的面積的最大值及此時對角線

與

的交點坐標.

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

的離心率為

,過右焦點

且斜率為

的直線與

相交于

兩點.若

,則

( )
A.1 | B. | C. | D.2 |
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:單選題
橢圓的一個頂點與兩個焦點構(gòu)成等邊三角形,則橢圓的離心率( )
查看答案和解析>>