圆锥曲线专题解析4:构造齐次式求解离心率(附参考答案)
圆锥曲线专题解析4:构造齐次式求解离心率
Ø方法导读
求椭圆或双曲线的离心率的值一直是高考中的热点,其所考题型涉及知识点较多,处理的思路和方法一般也比较灵活,因此不少同学觉得比较困难.其实,有关圆锥曲线的离心率求解问题并非杂乱无章、毫无规律,而是有章可循的,本文将就构造齐次式求解离心率的方法和技巧加以归纳总结.
Ø高考真题
【2019·全国Ⅰ卷理·16】已知双曲线
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的左、右焦点分别为F1,F2,过F1的直线与
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的两条渐近线分别交于
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,
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两点.若
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,
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,则
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的离心率为________.
Ø解题策略
要计算双曲线的离心率,首先画出示意图,得到
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的大概位置,然后根据
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,
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,推出
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是
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中点,且
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,接下来怎么利用这些条件求解离心率是每个学生头痛的问题.因为题目中没有具体的值,所以
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的值应该是不具体的,这时候我们就要想到构造
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的齐次式,利用比例关系来求解离心率.因为离心率是个比值,可以不用
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的具体值,知道他们的比例关系即可.怎么通过条件得到齐次式,中点和垂直关系怎么用,这是解决问题的关键.当我们无从下手的时候,咋办,就大胆的尝试,设出
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或
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的坐标,根据中点关系就可以得到另外一个点的坐标,然后由点在渐近线上和垂直关系建立两个等式,应该就能消去假设的未知数得到
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或其中两个的关系式,然后求解.这其实也就是构造齐次式的思路,利用条件消去假设的参数得到
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的比例关系或等式,然后就可以得到离心率.
Ø解题过程
解析:不妨设点
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,故
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,
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,由
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可得
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,解得
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,故
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,又
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,故
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,代入直线
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可得
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,解得
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,故离心率为
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.
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Ø解题分析
上述解法中根据点
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的位置,设出
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的坐标为
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,然后通过垂直关系可得到
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,从而解得
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,这样就消去了假设的未知数,再根据
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是
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中点,得到
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的坐标,结合
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在另一条渐近线上,代入直线方程得到一个等式关系,化简有
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,然后就可以解出离心率了.构造齐次式一般用在题目中没给出一些具体的值的条件,
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不能具体算出的场合,这个时候可以利用题目给出的条件化简得到
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的比例关系求解离心率.条件怎么用,这个要根据具体的题目具体分析,原则上就是通过条件消去其它的未知数,化简到最后式子中只含
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或其中两个的齐次式,然后求解即可.
Ø拓展推广
1.方程思想:齐次方程、不等式
(1)若给定椭圆或双曲线的方程,则根据方程确定
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,
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,进而求出
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,
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的值,然后利用公式
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直接求解;
(2)若椭圆或双曲线方程未知,则根据条件及几何图形建立关于
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,
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,
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的齐次等式(或不等式),然后化为关于
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,
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的齐次方程(或不等式),进而得到关于离心率
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的方程(或不等式)进行求解.
2.离心率相关
椭圆的离心率为
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,越大越扁;
双曲线的离心率为
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,越大开口越大;
抛物线的离心率为
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.
3.相关公式
3.1椭圆中
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,双曲线中
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;
3.2焦半径公式:
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分别为椭圆或双曲线的左右焦点,
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为曲线上任意一点,则
椭圆:
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,
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.
双曲线:
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,
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.
(双曲线焦半径符号口诀:绝对值内看焦点,左加右减,去绝对值看分支,左负右正)
4.焦点三角形:
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设
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分别为椭圆或双曲线的左右焦点,
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为曲线上任意一点,
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,
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,
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,则
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.
双曲线中的结论为:
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.
变式训练1
设椭圆
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的右焦点为
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,右顶点为
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,点
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,
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是椭圆
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上关于原点对称的两点(
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,
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均不在
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轴上),若直线
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与
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的交点平分线段
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,求
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的离心率.

变式训练2
已知
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,
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分别为椭圆

的左、右焦点,
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为椭圆上一点,
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为坐标原点,且
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,
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,求该椭圆的离心率.
变式训练3
设双曲线

的半焦距为

,直线

过

,

两点,已知原点到直线

的距离为

,求双曲线的离心率.
变式训练4
已知椭圆方程为
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
,

为椭圆的左顶点,

是右焦点,

是短轴的一个顶点,

,求椭圆的离心率.

变式训练5
设双曲线

的右焦点为

,过点
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作与

轴垂直的直线
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交两渐近线于

,
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两点,且与双曲线在第一象限的交点为

,设

为坐标原点,若

,

,求该双曲线的离心率.
答案
变式训练1
见解析
设

的中点为

,
依题意得点
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,
根据中点坐标公式可得
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,
∵点

,

关于原点对称,
∴

,
∵

,其中

.
由
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,

,
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三点共线得

,
∴

,
化简得

,
∴椭圆

的离心率为

.
变式训练2
见解析
∵

位于

,即

,
又

,
所以
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,
又
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,且

,
∴

,
∴

,
∴

,
∴

,即

.

变式训练3
见解析
由已知,设直线

的方程为

,
整理得

,
由点到直线的距离公式,得

,
又

,
∴

,
两边平方,
得

,
两边同时除以

,
整理得

,
即

,
解得

或

,
又

,
∴

,
∴

,
∴

,
故双曲线的离心率为

.

变式训练4
见解析
由题意知

,

,

,

,
∵

,
∴

,
即

,
整理得

,
两边同时除以

,
得

,
解得

,

(舍去),
∴椭圆的离心率为

.
变式训练5
见解析
∵双曲线的渐近线为

,焦点

,
∴

,

,

,
∵

,
∴

,
∴

,

,
解得

,

,
又由

,
得

,
解得

,
∴

,即双曲线的离心率为

.
