若双曲线的两条渐近线互相垂直，则其离心率为＿＿＿＿＿＿。 2 离心率渐近线垂直离心率等轴双曲线全国甲卷 2022

2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> 渐近线为 y = ± a b x ，两条斜率之积为 − a 2 b 2 。互相垂直要求斜率之积为 − 1 ： a 2 b 2 = 1 ，即 a = b 。代入 c 2 = a 2 + b 2 = 2 a 2 ，得 e = 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/> 。

高考数学考点题库/双曲线/离心率

2022 年全国甲卷·第15题

填空题★★★★☆ 较难改编题通法

若双曲线的两条渐近线互相垂直，则其离心率为＿＿＿＿＿＿。

答案

2

分步解析

1渐近线为 $y = \pm \frac{b}{a} x$ ，两条斜率之积为 $- \frac{b ^{2}}{a ^{2}}$ 。
2互相垂直要求斜率之积为 $- 1$ ： $\frac{b ^{2}}{a ^{2}} = 1$ ，即 $a = b$ 。
3代入 $c^{2} = a^{2} + b^{2} = 2 a^{2}$ ，得 $e = 2$ 。

#渐近线垂直#离心率#等轴双曲线

2022 年真题全国甲卷

想按考点连续练习？前往双曲线专题页查看全部真题与解析。

2022 年全国甲卷·第15题

分步解析

同考点推荐双曲线 · 离心率