Question

A hyperbola passes through the point $P(\sqrt{2},\sqrt{3})$ and has focii at $(\pm 2,0).$ Then the tangent to this hyperbola at $P$ also passes through the point:

• A. $(3\sqrt{2},2\sqrt{3})$
• B. $(2\sqrt{2},3\sqrt{3})$
• C. $(\sqrt{3},\sqrt{2})$
• D. $(-\sqrt{2},-\sqrt{3})$

Answer 1

The correct option is B $(2\sqrt{2},3\sqrt{3})$

Coordinates of foci are $(\pm 2,0)$
$\therefore ae=2\Rightarrow a^2{e}^2=4\Rightarrow a^2+b^2=4\Rightarrow b^2=4-a^2$
Thus, equation of hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{4-a^2}=1$
Given, the hyperbola passes through $P(\sqrt{2},\sqrt{3})$,
$\Rightarrow \frac{2}{a^2}-\frac{3}{4-a^2}=1\Rightarrow a^4-9a+8=0\Rightarrow a^2=1,8[\text{Ignoring}{a}^2=8\text{as for}{a}^2=8\Rightarrow b^2=-4]\therefore a^2=1$

$\therefore$ Equation of hyperbola is $\frac{x^2}{1}-\frac{y^2}{3}=1$
Hence equation of tangent at $P$ is
$\sqrt{2}x-\frac{\sqrt{3}y}{3}=1\Rightarrow \sqrt{6}x-y-\sqrt{3}=0$
From the given options only $(2\sqrt{2},3\sqrt{3})$ is correct.