Find the parametric form of normal for the ellipse $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$?

Question

$\begin{array}{1 1}(a)\;\large\frac{ax}{\cos\theta}-\frac{by}{\sin \theta}\normalsize =a^2-b^2\\(b)\;\large\frac{ax}{\cos\theta}-\frac{by}{\sin \theta}\normalsize =a^2+b^2\\(c)\;\large\frac{ax}{\sin\theta}-\frac{by}{\cos \theta}\normalsize =a^2-b^2\\(d)\;\text{None of these}\end{array}$

Answer 1

Equation of normal at $(x_1,y_1)$ is

$\large\frac{a^2x}{x_1}-\frac{b^2y}{y_1}$$=a^2-b^2$

Parametric coordinate $(x_1,y_1)$ is $(a\cos\theta,b\sin\theta)$

Hence $\large\frac{a^2x}{a\cos \theta}-\frac{b^2y}{b\sin \theta}$$=a^2-b^2$

$\large\frac{ax}{\cos\theta}-\frac{by}{\sin \theta}\normalsize =a^2-b^2$

Hence (a) is the correct answer.