Question

Observe the following statements
I: If $p$ and $q$ are the lengths of perpendiculars from the origin on the tangent and normal at any point on the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$ then $4p^2+q^2=1$.
II: If the tangent at any point $P$ on the curve $x^3.y^2=a^5$ cuts the coordinate axes at $A$ and $B$ then $AP:PB=3:2$

• A. only I
• B. only II
• C. both I and II
• D. neither I nor II

Answer 1

The correct option is A only I

$x=a{\cos }^3\theta$
$y=a{\sin }^3\theta$
$y^{\prime }=-\tan \theta$
$(y-si{n}^3\theta )=-\tan \theta (x-{\cos }^3\theta )$
$y+n\tan \theta -\tan \theta {\cos }^3\theta -{\sin }^3\theta =0$
Perpendicular from origin

$\left|\frac{\tan \theta {\cos }^3\theta +{\sin }^3\theta }{\sqrt{1+{\tan }^2\theta }}\right|=p$
$p_2\sin \theta \cos \theta$
$p=\frac{{\sin }^2\theta }{2}$
$(y-\sin 3\theta )=\frac{1}{\tan \theta }(x-\cos 3\theta )$
$y\tan \theta -\sin 3\theta \tan \theta -x+\cos 3\theta =0$
Perpendicular frm the origin
$\left|\frac{\cos 3\theta -\sin 3\theta \tan \theta }{\sqrt{1+{\tan }^2\theta }}\right|=q$
$q=\cos 2\theta$
$4p^2+q^2=1$