Question

If a hyperbola passes through the point P $(\sqrt{2},\sqrt{3})$ and has foci $(\pm 2,0),$ then the tangent to this hyperbola at P also passes through the point

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

Answer 1

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

Let the equation of hyperbola be $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
$\therefore ae=2\Rightarrow a^2{e}^2=4\Rightarrow a^2+b^2=4\Rightarrow b^2=4-a^2\therefore \frac{x^2}{a^2}-\frac{y^2}{4-a^2}=1Since,(\sqrt{2},\sqrt{3})lieonhyperbola.\therefore \frac{2}{a^2}-\frac{3}{4-a^2}=1\Rightarrow 8-2a^2-3a^2=a^2(4-a^2)\Rightarrow 8-5a^2=4a^2-a^4\Rightarrow a^4-9a^2+8=0\Rightarrow (a^2-8)(a^2-1)=0\Rightarrow a^2=8,a^2=1$
Now equation of hyperbola is $\frac{x^2}{1}-\frac{y^2}{3}=1$
$\therefore$ equation of tangent at $(\sqrt{2},\sqrt{3})isgivenby\sqrt{2}x-\frac{\sqrt{3}y}{3}=1\Rightarrow \sqrt{2}x-\frac{y}{\sqrt{3}}=1$
which passes through the point $(2\sqrt{2},3\sqrt{3}).$