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  4. If the line x-1=0, is a directrix of the hyperbola kx 2- y 2=6, then the hyperbola passes through the point

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Q. If the line $x-1=0$, is a directrix of the hyperbola $kx ^2- y ^2=6$, then the hyperbola passes through the point

JEE MainJEE Main 2022Conic Sections

Solution:

$\frac{x^2}{6 / k}-\frac{y^2}{6}=1 .....$(1)
$ e^2=1+\frac{6}{6 / k}$
$e=\sqrt{1+k} $
$ a=\sqrt{\frac{6}{k}}$
Eq. of directrix $x=\frac{a}{e} \Rightarrow x=\sqrt{\frac{6}{k(k+1)}}$
$\frac{6}{ k ( k +1)}=1$
$k =2$
From eq. (1), we get $2 x^2-y^2=6$
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