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

$(ae{)}^2=4\Rightarrow b^2=4-a^2$
Thus, equation of hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{4-a^2}=1$
Given, the hyperbola passes through $P(\sqrt{2},\sqrt{3}).$
$\Rightarrow \frac{2}{a^2}-\frac{3}{4-a^2}=1$
$\Rightarrow a^2=1,8[\text{Ignoring}{a}^2=8\text{as}{a}^2<(ae{)}^2]$
$\Rightarrow a^2=1$
$\therefore$ Equation of hyperbola is $\frac{x^2}{1}-\frac{y^2}{3}=1$
Slope of tangent at $P(\sqrt{2},\sqrt{3})$ is $\frac{dy}{dx}{|}_{(\sqrt{2},\sqrt{3})}=\sqrt{6}$
$\therefore$ Equation of tangent at $P(\sqrt{2},\sqrt{3})$ is $y-\sqrt{3}=\sqrt{6}(x-\sqrt{2})$
Hence, the tangent passes through $(2\sqrt{2},3\sqrt{3}).$