Question

If the chords of tangents from two points $(–4,2)$ and $(2,1)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are at right angle, then the eccentricity of the hyperbola is

• A. $\sqrt{2}$
• B. $\frac{\sqrt{7}}{2}$
• C. $\sqrt{\frac{3}{2}}$
• D. $\sqrt{\frac{5}{3}}$

Answer 1

The correct option is C $\sqrt{\frac{3}{2}}$

Equation of chord of contact with respect to point $(–4,2)$ is $-\frac{4x}{a^2}-\frac{2y}{b^2}=1$ and with respect to point $(2,1)$ is $\frac{2x}{a^2}-\frac{y}{b^2}=1$
Now, according to the given condition
$\left(\frac{\left(\frac{4}{a^2}\right)}{\left(\frac{-2}{b^2}\right)}\right)\times \frac{\left(\frac{-2}{a^2}\right)}{(-\frac{1}{b^2})}=-1$
$\Rightarrow \frac{b^4}{a^4}=\frac{1}{4}\Rightarrow \frac{b^2}{a^2}=\frac{1}{2}$
$e=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{\frac{3}{2}}$