Question

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

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

Answer 1

Answer: (A)

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
$a^2+b^2=4$
and $\frac{2}{a^2}-\frac{3}{b^2}=1$
$\frac{2}{4-b^2}-\frac{3}{b^2}=1$
$\Rightarrow b^2-3$
$\therefore a^2=1$
$\therefore x^2-\frac{y^2}{3}=1$
$\therefore$ Tangent at $P\left(\sqrt{2},\sqrt{3}\right)$ is $\sqrt{2}x-\frac{y}{\sqrt{3}}=1$
Clearly it passes through $\left(2\sqrt{2},3\sqrt{3}\right)$