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Equation of Normal at Given Point

Assertion :ST...

Question

Assertion :STATEMENT-1 :Let $P$ be any point on a directrix of an ellipse.Then chords of contact of point $P$ with respect to the ellipse and its auxiliary circle intersect at corresponding focus Reason: STATEMENT-2 : The equation of family of lines passing through point of intersection of lines $L_{1} = 0$ and $L_{2} = 0$ is $L_{1} + λ L_{2} = 0$

Statement -1 is True, Statement -2 is True; Statement -2 is a correct explanation for Statement -1

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Statement-1 is True, Statement-2 is True ;Statement-2 is NOT a correct explanation for Statement-1

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Statement -1 is True, Statement -2 is False

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Statement -1 is False, Statement -2 is True

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Solution

The correct option is A
Statement -1 is True, Statement -2 is True; Statement -2 is a correct explanation for Statement -1

Let ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ where $a > b > 0$ and point P is
$(\frac{a}{e}, k)$
Auxiliary circle is $x^{2} + y^{2} = a^{2}$
Chords of contact are $\frac{x a}{e a^{2}} + \frac{y k}{b^{2}} = 1$ and $\frac{x a}{e} + y k = a^{2}$
$\Rightarrow (\frac{x}{a e} - 1) + k (\frac{y}{b^{2}}) = 0$ and $(\frac{x a}{e} - a^{2}) + k y = 0$
which are family of concurrent lines passing through point $(a e, 0)$ i.e. focus of ellipse

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