题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133637-92272.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133638-22768.png)
(1)求动点Q的轨迹
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133638-21291.png)
(2)已知A,B,C是轨迹
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133638-21291.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133638-43003.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133638-89624.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-56261.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-69630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-23899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-74848.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133640-51037.png)
解析
试题分析:(1)连结QF,由于线段的垂直平分线上的点到线段两端点的距离相等,所以|QE|+|QF|=|QE|+|QP|=4
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133640-27899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133638-21291.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133640-70563.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133640-42566.png)
(1)连结QF,根据题意,|QP|=|QF|,则|QE|+|QF|=|QE|+|QP|=4
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133640-27899.png)
故动点Q的轨迹
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133638-21291.png)
设其方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133641-92070.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133641-74384.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133641-79094.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133641-82426.png)
所以点Q的轨迹
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133638-21291.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-56261.png)
(2)存在最小值. 5分
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133642-22345.png)
(ⅰ)当AB为长轴(或短轴)时,可知点C就是椭圆的上、下顶点(或左、右顶点),则
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133640-70563.png)
(ⅱ)方法一、当直线AB的斜率存在且不为0时,设斜率为k,则直线AB的直线方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133640-42566.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133642-46710.png)
联立方程组
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133642-99555.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133642-88563.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133642-70776.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133638-43003.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133643-61056.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133643-19114.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133643-27321.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133643-71351.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133644-43034.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133644-82071.png)
由于
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133644-77305.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133644-95222.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-13646.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-69577.png)
综合(ⅰ)(ⅱ),当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-69577.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-17990.png)
此时
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-24938.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133646-12581.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133646-97614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133646-71200.png)
所以点C的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-69630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-23899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-74848.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133640-51037.png)
方法二、前同(ⅰ),记
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133646-72211.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133646-21514.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133647-46049.png)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133647-23249.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133647-39413.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-69577.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133647-16612.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133648-55815.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133648-31483.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-17990.png)
综合(ⅰ)(ⅱ),当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-69577.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-17990.png)
此时
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-24938.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133646-12581.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133646-97614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133646-71200.png)
所以点C的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-69630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-23899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-74848.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133640-51037.png)
方法三、设
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133648-80997.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133648-34910.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133648-13910.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133649-93342.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133638-43003.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133649-71620.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133649-13465.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133649-81905.png)
且点C在椭圆上,则
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133650-42397.png)
联立①②,解得
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133650-64522.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133650-91020.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133650-59718.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133651-35873.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133651-34055.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133651-41789.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133651-13179.png)
记
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133652-29277.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133652-45436.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133652-72756.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133652-37267.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133652-60374.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133653-75338.png)
综合(ⅰ)(ⅱ),当
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133653-75338.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133648-31483.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133645-17990.png)
所以点C的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-69630.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-23899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133639-74848.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133640-51037.png)
核心考点
试题【如图,已知圆E:,点,P是圆E上任意一点.线段PF的垂直平分线和半径PE相交于Q.(1)求动点Q的轨迹的方程;(2)已知A,B,C是轨迹的三个动点,A与B关于原】;主要考察你对椭圆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/20191023/20191023133627-35799.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133627-67036.png)
(1)求动点Q的轨迹
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133627-97074.png)
(2)点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133628-74649.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133628-42236.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133627-97074.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133628-81210.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133628-24897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133620-93969.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133620-94187.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133620-76083.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133620-53722.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133621-25890.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133621-57969.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133621-67774.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133620-53722.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133621-56788.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133622-13172.png)
A.0 | B.1 | C.![]() | D.2 |
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133547-62301.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133547-17946.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133548-45074.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133548-47628.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133548-96494.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133548-69356.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133549-68491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133549-94461.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133549-65436.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133549-74363.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133548-47628.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133550-53301.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133550-97501.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133550-34722.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133550-50072.png)
(1)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133548-45074.png)
(2)若点
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133551-94648.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133551-56994.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133548-47628.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133551-56994.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133541-76768.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133541-10738.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133541-86838.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133542-61752.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133541-86838.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133542-85896.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191023/20191023133539-78131.png)
A.16 B.11 C.8 D.3
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