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 points:

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

Answer 1

Answer: (C)

$(2\sqrt{2},3\sqrt{3})$

$\Rightarrow$ The equation of hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\Rightarrow$ foci is $(\pm 2,0)$, so $ae=2\Rightarrow a^2{e}^2=4$

$b^2=a^2(e^2-1)$

$\therefore$ $a^2+b^2=4$ ----- ( 1 )

$\Rightarrow$ Hyperbola passes through the pointv$\sqrt{2},\sqrt{3}$

$\therefore$ $\frac{2}{a^2}-\frac{3}{b^2}=1$ ------ ( 2 )

On solving ( 1 ) and ( 2 ), we get

$a^2=8$ ( is rejected ) and $a^2=1$ and $b^2=3$

$\therefore$ $\frac{x^2}{1}-\frac{y^2}{3}=1$

$\Rightarrow$ Equation of tangent is $\frac{\sqrt{2}x}{1}-\frac{\sqrt{3}y}{3}=1$

Hence, $P$ also passes thrugh the points $(2\sqrt{2},3\sqrt{3})$