若双曲线 x 2 a 2 - y 2 b 2 =1 的一条渐近线为 y=2x ，则其离心率为＿＿＿＿＿＿。 5 离心率离心率渐近线双曲线新课标I卷 2023

5 <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"/> 由渐近线 y = a b x = 2 x 得 a b = 2 ，即 b = 2 a 。代入 c 2 = a 2 + b 2 = a 2 + 4 a 2 = 5 a 2 。故 e = a c = 5 <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

2023 年新课标I卷·第13题

填空题★★★☆☆ 中档改编题通法

若双曲线

\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1

的一条渐近线为

y = 2 x

，则其离心率为＿＿＿＿＿＿。

答案

5

分步解析

1由渐近线 $y = \frac{b}{a} x = 2 x$ 得 $\frac{b}{a} = 2$ ，即 $b = 2 a$ 。
2代入 $c^{2} = a^{2} + b^{2} = a^{2} + 4 a^{2} = 5 a^{2}$ 。
3故 $e = \frac{c}{a} = 5$ 。

点评双曲线的渐近线斜率与离心率由同一个量

\frac{b}{a}

决定：

e = 1 + (\frac{b}{a})^{2}

，知一即可求另一。

#离心率#渐近线#双曲线

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2023 年新课标I卷·第13题

分步解析

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