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  4. Let x=4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is (1/2). If P (1, β), β>0 is a point on this ellipse, then the equation of the normal to it at P is :-

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Q. Let $x=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac{1}{2}$. If $P (1, \beta), \beta>0$ is a point on this ellipse, then the equation of the normal to it at $P$ is :-

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Solution:

Ellipse : $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
$\text { directrix }: x=\frac{a}{e}=4 \& e=\frac{1}{2} $
$\Rightarrow a=2 \& b^{2}=a^{2}\left(1-e^{2}\right)=3$
$\Rightarrow $ Ellipse is $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$
$P$ is $\left(1, \frac{3}{2}\right)$
Normal is: $\frac{4 x}{1}-\frac{3 y}{3 / 2}=4-3$
$\Rightarrow 4 x-2 y=1$