Q. If $x\cos{\alpha + y\sin\alpha =}p$ is a tangent to the ellipse, then
A) $a^{2}\sin^{2}\alpha + b^{2}\cos^{2}\alpha = p^{2}$
B) $a^{2} + b^{2}\sin^{2}\alpha = p^{2}cosec^{2}\alpha$
C) $a^{2}\cos^{2}\alpha + b^{2}\sin^{2}\alpha = p^{2}$
D) None of the above
Explanation
(c)
We have, $x\cos{\alpha + y\sin\alpha =}p$
$\Rightarrow y = - x\ \cot{\alpha +}p\ cosec\ \alpha$
Since, above line is tangent to the ellipse
$\therefore c^{2} = a^{2}m^{2} + b^{2}$
$\Rightarrow p^{2}coses^{2}\alpha = a^{2}\cot^{2}\alpha + b^{2}$
$\Rightarrow a^{2}\cos^{2}\alpha + b^{2}\sin^{2}\alpha = p^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.