Q. If the chords of constant of tangents from two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ 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) $- \frac{a^{4}}{b^{4}}$
D) $- \frac{b^{4}}{a^{4}}$
Explanation
(c)
Chord of contact are
$\frac{xx_{1}}{a^{2}} + \frac{yy_{1}}{b^{2}} = 1$ and $\frac{xx_{2}}{a^{2}} + \frac{yy_{2\ }}{b^{2}} = 1$
Product of slopes $= - 1$
$\Rightarrow \left( - \frac{x_{1}}{a^{2}} \bullet \frac{b^{2}}{y_{1}} \right)\left( - \frac{x_{2}}{a^{2}} \bullet \frac{b^{2}}{y_{2}} \right) = - 1$
$\Rightarrow \frac{x_{1}x_{2}}{y_{1}y_{2}} = - \frac{a^{4}}{b^{4}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.