Question

If the line x – 1 = 0 is a directrix of the hyperbola kx2 – y2 = 6, then the hyperbola passes through the point

• A. $(-2\sqrt5,6)$
• B. $(-\sqrt5,3)$
• C. $(\sqrt5,-2)$
• D. $(2\sqrt5,3\sqrt6)$

Answer 1

Answer: (C)

$(\sqrt5,-2)$

Answer 2

Given hyperbola : $\frac{x^2}{6/k}-\frac{y^2}{6} = 1$
$e = \sqrt{1+\frac{6}{6/k}}=\sqrt{1+k}$
$x = ± \frac{a}{e} ⇒ x = ± \frac{\sqrt6}{\sqrt{k}\sqrt{k+1}}$
As given : $\frac{\sqrt6}{\sqrt{k}\sqrt{k+1}}=1$
$⇒ k = 2$
$⇒ \frac{x^2}{3}-\frac{y^2}{6} = 1$
Hence, the option that satisfies and is the correct option is (C):$(\sqrt{5},-2)$