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# Find the parametric form of normal for the circle $x^2+y^2=a^2$

$\begin{array}{1 1}(a)\;\large\frac{y}{\sin\theta}=\frac{x}{\cos\theta}\\(b)\;\large\frac{x}{\sin\theta}=\frac{y}{\cos\theta}\\(c)\;\large\frac{y}{\sin^2\theta}=\frac{x}{\cos^2\theta}\\(d)\;\text{None of these}\end{array}$

Parametric co-ordinates are ($a\cos \theta,a\sin\theta)$
Slope of tangent at ($a\cos \theta,a\sin\theta)$
$2x+2yy'=0$
$2\times a\cos\theta+2\times a\sin \theta y'=0$
$y'=-\large\frac{\cos\theta}{\sin\theta}$
Hence slope of normal is $\large\frac{\sin \theta}{\cos\theta}$
Equation of normal is $(y-a\cos\theta)=\large\frac{\sin \theta}{\cos\theta}$$(x-a\sin \theta)$
$\large\frac{y}{\sin\theta}=\frac{x}{\cos\theta}$
Hence (a) is the correct answer.