First slide

Ellipse

Question

If chords of contact of the tangent 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 $\frac{x_{1} x_{2}}{a^{2}} \times \frac{y_{1} y_{2}}{b^{2}}$ is equal to

Moderate

A

$\frac{a^{2}}{b^{2}}$
B

$\frac{- b^{2}}{a^{2}}$
C

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

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

Solution

Chords of contacts are $\frac{x x_{1}}{a^{2}} + \frac{y y_{1}}{b^{2}} = 1$ and $\frac{x x_{2}}{a^{2}} + \frac{y y_{2}}{b^{2}} = 1$
Since they are at right angles
$\begin{array}{l} \frac{- b^{2}}{a^{2}} \times \frac{x_{2}}{y_{2}} \times \frac{- b^{2}}{a^{2}} \times \frac{x_{1}}{y_{1}} = - 1 \\ \Rightarrow \frac{x_{1} x_{2}}{y_{1} y_{2}} = \frac{- a^{4}}{b^{4}} \end{array}$

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