If $y = mx - \frac{(a^{2} - b^{2})m}{\sqrt{a^{2} + b^{2}m^{2}}}$ is normal to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ for all values of $m$ belonging to

Question

Dr. Abubacker · Accepted Answer

(c)
The equation of normal to the given ellipse at $P(a\cos	heta,\ b\sin	heta)$ is
$ax\sec{	heta - by\ cosec\ 	heta - a^{2} = b^{2}}$
$\Rightarrow y = \left( \frac{a}{b}	an	heta ight)x - \frac{a^{2} - b^{2}}{b}\sin	heta$ …(i)
Let $\frac{a}{b}	an{	heta = m}$, then $\sin{	heta = \frac{bm}{\sqrt{a^{2} + b^{2}m^{2}}}}$
$	herefore$ From Eq. (i), we get
$y = mx - \frac{(a^{2} - b^{2})m}{\sqrt{a^{2} + b^{2}m^{2}}}$
$\because\frac{a}{b}	an{	heta \in R \Rightarrow m \in R}$

Q. If $y = mx - \frac{(a^{2} - b^{2})m}{\sqrt{a^{2} + b^{2}m^{2}}}$ is normal to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ for all values of $m$ belonging to

Explanation

Reviewed by Dr. Abubacker

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