Question

lf the normal at the point $p\left(\theta \right)$ of the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ passes through the origin then

• A. $\theta =\frac{\pi }{3}$
• B. $\theta =\frac{\pi }{6}$
• C. $\theta =\frac{\pi }{4}$
• D. $\theta =\frac{\pi }{2}$

Answer 1

The correct option is C $\theta =\frac{\pi }{4}$

$x=a{\cos }^3\theta$
$y=a{\sin }^3\theta$
$y^1=\frac{a3{\sin }^2\theta \cos \theta }{a3{\cos }^2\theta -\cos \theta }$
$\frac{-\sin \theta }{\cos \theta }(y-a{\sin }^3\theta )=\frac{\cos \theta }{\sin \theta }(x-a{\cos }^3\theta )$
$(0,0)$ satisfies the eq${}^n$
${\sin }^4\theta ={\cos }^4\theta$
$\theta =\frac{\pi }{4}$