题目
题型:不详难度:来源:
已知抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-67021.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-91050.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-91050.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-24887.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-16795.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-99688.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-86629.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-56549.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-91050.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-23956.gif)
(1)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-86629.gif)
(2)经过
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-16795.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-99688.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-43566.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181654-43444.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-43566.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181654-43444.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181654-96617.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-45851.gif)
(3)椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-86629.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-80611.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-80611.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-18488.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-49006.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181656-44382.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181656-26552.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181656-30504.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-91050.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-18488.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-49006.gif)
![]() |
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181657-87476.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181657-80548.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181657-91353.jpg)
(考查椭圆、抛物线、直线、定积分等知识,考查数形结合、化归转化等数学思想、以及推理论证能力和运算求解能力)
解:(1)设椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-86629.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181658-36287.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181658-45742.gif)
由已知条件,得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181658-98512.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181658-57410.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181659-66134.gif)
所以椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-86629.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181657-87476.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181659-74382.gif)
(2)显然直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181659-23464.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181659-23464.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
故可设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181659-23464.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181700-45673.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181700-57396.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181700-80485.gif)
消去
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181701-36724.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181701-72283.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181701-87281.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181701-36412.gif)
∵抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181702-58913.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181702-43830.gif)
∴过抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-16795.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181652-99688.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181702-50368.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181703-92479.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181703-77275.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181703-46986.gif)
解得两条切线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181703-38330.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181704-57886.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181654-96617.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181704-55475.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181704-77337.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181705-92269.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181705-72027.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181705-97841.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181706-41525.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-45851.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181706-59504.gif)
(3)假设存在点
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-80611.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-80611.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181706-78390.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181706-78390.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-86629.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-80611.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181706-32878.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181657-91353.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-80611.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181707-85791.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181707-24132.gif)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181707-31390.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181707-43201.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181708-80266.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181708-95773.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181709-56665.gif)
故不妨取
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181709-25940.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181709-53169.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181710-37151.gif)
综上所述,椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181653-86629.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181706-32878.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181655-80611.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181710-52664.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181711-25452.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181656-44382.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181656-26552.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181709-53169.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181710-37151.gif)
此时,两切线的方程分别为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181711-24432.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181712-79076.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181712-74366.gif)
抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181651-83014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181710-52664.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181711-25452.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181657-80548.gif)
核心考点
试题【20.(本小题满分14分)已知抛物线:的焦点为,过点作直线交抛物线于、两点;椭圆的中心在原点,焦点在轴上,点是它的一个顶点,且其离心率.(1)求椭圆的方程;(2】;主要考察你对抛物线的定义与方程等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181618-60380.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181618-69415.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181618-81246.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181619-84128.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181619-44785.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181619-74298.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181619-39582.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181619-39582.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181619-89063.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181600-11730.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181600-55672.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181600-29771.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181600-91837.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181600-11730.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181600-12118.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181601-39575.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181600-12118.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181601-39575.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181601-80693.gif)
(1)求抛物线的标准方程;
(2)求
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181601-43024.gif)
(3)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181601-79005.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181601-17476.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181601-84780.gif)
如图,已知抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-22527.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-33568.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-82094.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-33568.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-82094.gif)
线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-25251.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-32438.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-49865.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-32438.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-49865.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-33568.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181541-45033.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181541-53550.gif)
(1)求证:直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181541-97391.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181541-93925.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181541-40944.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181541-40944.gif)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181541-72724.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181541-24845.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181541-16671.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181540-82094.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181542-50806.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181534-90515.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181535-79403.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181535-16185.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181535-45817.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181535-36326.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181535-71001.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181535-72689.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181536-84033.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181536-34226.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181528-32466.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181528-70420.gif)
A.
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181528-46524.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181528-74496.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181529-65950.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025181529-73088.gif)
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