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

Answer: (D)

$-\frac{a^4}{b^4}$

Answer 2

Chords of contact are $\frac{xx_{1}}{a^{2}}-\frac{yy_{1}}{b^{2}} = 1, \frac{xx_{2}}{a^{2}}-\frac{yy_{2}}{b^{2}} = 1$ These are at right angles. $\therefore\frac{b^{2}}{a^{2}}\cdot\frac{x_{1}}{y_{1}}\cdot\frac{b^{2}}{a^{2}}\cdot\frac{x_{2}}{y_{2}} = -1$ $\therefore \frac{x_{1}x_{2}}{y_{1}y_{2}} = -\frac{a^{4}}{b^{4}}$