Let H be the hyperbola, whose foci are ( 1 2 , 0 ) and eccentricity is 2 . Then the length of its latus rectum is ___________ . [2023]

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Solution

Q.
Let H be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$ . Then the length of its latus rectum is ___________ . [2023]

1 $3$

2 $\frac{5}{2}$

3 $2$

4 $\frac{3}{2}$

Ans.
(3)

$Distance between two foci = 2 a e = 2 \sqrt{2}$

$\Rightarrow a e = \sqrt{2}$ $\Rightarrow a (\sqrt{2}) = \sqrt{2} (∵ e = \sqrt{2})$

$\Rightarrow a = 1$

$We know that e^{2} = 1 + \frac{b^{2}}{a^{2}} \Rightarrow 2 = 1 + \frac{b^{2}}{1}$

$\Rightarrow b^{2} = 1 \Rightarrow b = 1$

$∴$ $Length of latus rectum = \frac{2 b^{2}}{a} = \frac{2 {(1)}^{2}}{1} = 2$

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