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Q.
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 :
$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
$a^{2}+b^{2}=4$
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 $\left(2\sqrt{2},3\sqrt{3}\right)$