A chord PQ of the ellipse frac{x^2}{9} +&nbsp | vedclass

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Question

Let the point of intersection be $\mathrm{R}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right) .$ Then $\mathrm{PQ}$ is the chord of contact of the ellipse w

Vedclass · Accepted Answer

an ellipse

Vedclass · Answer

Let the point of intersection be $\mathrm{R}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right) .$ Then $\mathrm{PQ}$ is the chord of contact of the ellipse with respect to $\mathrm{R}$ and its equation will be $\frac{\mathrm{xx}_{1}}{9}+\frac{\mathrm{yy}_{1}}{4}=1$

Now combined equation of lines joining $P, Q$ to centre $\mathrm{O}(0,0)$ is [It is obtained making $\mathrm{x}^{2} / 9+\mathrm{y}^{2} / 4=1$homogeneous with help of $(1)]$

$\frac{x^{2}}{9}+\frac{y^{2}}{4}=\left(\frac{x x_{1}}{9}+\frac{y y_{1}}{4}-1\right)^{2}$

As given $OP$ $\perp$ $OQ,$ so coefficient of

$\mathrm{x}^{2}+$ coefficient of $\mathrm{y}^{2}=0$

$\Rightarrow\left(\frac{x_{1}^{2}}{81}-\frac{1}{9}\right)+\left(\frac{y^{2}}{16}-\frac{1}{4}\right)=0$

Hence $\operatorname{locus}$ of $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ is $\frac{\mathrm{x}^{2}}{81}+\frac{\mathrm{y}^{2}}{16}=\frac{13}{36},$ which is an ellipse.

A chord $PQ$ of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ subtends right angle at its centre. The locus of the point of intersection of tangents drawn at $P$ and $Q$ is-

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