Question

If the chords of contact of tangents from two points $(x_1,y_1)$ and $(x_2,y_2)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are at right angles, then $\frac{x_1{x}_2}{y_1{y}_2}$ is equal to

• A. $-\frac{a^2}{b^2}$
• B. $-\frac{b^2}{a^2}$
• C. $-\frac{b^4}{a^4}$
• D. $-\frac{a^4}{b^4}$

Answer 1

The correct option is D $-\frac{a^4}{b^4}$

Equation of chord of contact from $(x_1,y_1)$ and $(x_2,y_2)$ to the hyperbola are given by,
$\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1$ and $\frac{x{x}_2}{a^2}-\frac{y{y}_2}{b^2}=1$
Therefore slopes of these lines are $m_1=\frac{b^2}{a^2}.\frac{x_1}{y_1}$ and $m_2=\frac{b^2}{a^2}.\frac{x_2}{y_2}$
Given both lines are perpendicular
$\Rightarrow m_1\cdot m_2=-1\Rightarrow \frac{x_1{x}_2}{y_1{y}_2}=-\frac{a^4}{b^4}$
Hence, option 'D' is correct.