题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213714-35916.png)
(1)求抛物线C的方程;
(2)是否存在点M,使得直线MQ与抛物线C相切于点M?若存在,求出点M的坐标;若不存在,说明理由.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213714-48650.png)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213714-20119.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213715-33937.png)
因为抛物线C的准线方程为y=-
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213714-20119.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213715-78725.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213714-35916.png)
因此抛物线C的方程为x2=2y.
(2)假设存在点M(x0,
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213715-26065.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213715-49457.png)
所以直线MQ的方程为y-
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213715-26065.png)
令y=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-62783.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-52809.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-82467.png)
所以Q(
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-52809.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-82467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-62783.png)
又|QM|=|OQ|,
故(
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-82467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-52809.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-62783.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213715-26065.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-82467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-52809.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-75448.png)
因此(
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213716-62783.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213715-26065.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213717-51489.png)
又x0>0,所以x0=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213714-48650.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213714-48650.png)
故存在点M(
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213714-48650.png)
核心考点
试题【在平面直角坐标系xOy中,F是抛物线C:x2=2py(p>0)的焦点,M是抛物线C上位于第一象限内的任意一点,过M,F,O三点的圆的圆心为Q,点Q到抛物线】;主要考察你对曲线与方程的应用等知识点的理解。[详细]
举一反三
(Ⅰ)求抛物线C的方程;
(Ⅱ)过F作直线交抛物线于A、B两点.若直线OA、OB分别交直线l:y=x﹣2于M、N两点,求|MN|的最小值.
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213704-56059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213655-62515.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213655-23417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213655-92197.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213655-37159.png)
A.![]() ![]() | B.![]() ![]() | C.![]() ![]() | D.![]() ![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213635-45595.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213636-25594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213636-84828.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213636-64073.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213636-20981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213637-82383.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213637-36341.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213637-47952.png)
(I)求抛物线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213636-64073.png)
(2)现给出以下三个论断:①直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213638-97768.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213637-82383.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213638-62478.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213638-75071.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213639-63790.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213639-24135.png)
请你以其中的两个论断作为条件,余下的一个论断作为结论,写出一个正确的命题,并加以证明.
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213631-29106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213631-63483.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213617-29460.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213617-74029.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213617-14928.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213617-10093.png)
(1)求抛物线的方程,并写出焦点坐标;
(2)是否存在过焦点的直线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213617-35837.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213618-12397.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213618-27933.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213618-44772.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213618-24890.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025213617-35837.png)
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