If normal at any point P on ellipse frac{{{x^2}}}{{{a^2}}} + frac{{{y^2}}}{{{b^2}}} = 1 meets co-ord

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10-2. Parabola, Ellipse, Hyperbola

hard

If the normal at any point $P$ on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ meets the co-ordinate axes in $G$ and $g$ respectively, then $PG:Pg = $

A

$a:b$

B

${a^2}:{b^2}$

C

${b^2}:{a^2}$

D

$b:a$

Solution

(c) Let $P$ $(a\cos \theta ,\,b\sin \theta )$ be a point on the ellipse

$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1.$ Then the equation of the normal at $P$ is

$ax\sec \theta – by\,\,{\rm{cosec}}\theta = {a^2} – {b^2}$.

It meets the co-ordinate axes at $G\,\left( {\frac{{{a^2} – {b^2}}}{a}\cos \theta ,\,0} \right)$ and

$g\,{\rm{ }}\left( {0, – \frac{{{a^2} – {b^2}}}{b}\sin \theta } \right)$.

$ \Rightarrow P{G^2} = {\left( {a\cos \theta – \frac{{{a^2} – {b^2}}}{a}\cos \theta } \right)^2} + {b^2}{\sin ^2}\theta $

$ = \frac{{{b^2}}}{{{a^2}}}({b^2}{\cos ^2}\theta + {a^2}{\sin ^2}\theta )$

and $P{g^2} = \frac{{{a^2}}}{{{b^2}}}({b^2}{\cos ^2}\theta + {a^2}{\sin ^2}\theta )$, $\therefore$ $PG:Pg = {b^2}:{a^2}$.

Standard 11

Mathematics

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