Question

The locus of a variable point whose chord of contact w.r.t. the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ subtends a right angle at the origin is

• A. $\dfrac{x^2}{a^2} + \dfrac{y^2}{4b^2} = 1$
• B. $\left( \dfrac{x^2}{a^2} \right)^2 - \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$
• C. $\dfrac{x}{a} - \dfrac{y}{b} = \dfrac{1}{a^2} + \dfrac{1}{b^2}$
• D. $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = \dfrac{1}{2}$

Hint:

Use the condition for perpendicular tangents from a point to a conic to derive locus.

Answer 1

The Correct Option is D

Chord of contact subtends right angle at origin implies angle between tangents is $90^\circ$
Let $T = 0$ be the chord of contact. For conic, if tangents from a point $(x_1, y_1)$ are perpendicular, then:
$S_1 = \dfrac{1}{2} S$ where $S$ is the equation of conic and $S_1$ its value at point
For hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, condition becomes:
$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = \dfrac{1}{2}$