Question

The tangent at any point on the curve $x=a{\cos }^3\theta ,y=a{\sin }^3\theta$ meets the coordinate axes at $P$ and $Q$.
The locus of the mid point of $PQ$ is

• A. $x^2+y^2=a^2$
• B. $2(x^2+y^2)=a^2$
• C. $4(x^2+y^2)=a^2$
• D. $(x^2+y^2)=4a^2$

Answer 1

The correct option is C $4(x^2+y^2)=a^2$

$\frac{dy}{dx}=-\tan \theta$
Equation of tangent is
$y-a{\sin }^3\theta =-\tan \theta (x-a{\cos }^3\theta )$
$\frac{y}{\sin \theta }-a{\sin }^2\theta =-\frac{x}{\cos \theta }+a{\cos }^2\theta$
$\frac{x}{\cos \theta }+\frac{y}{\sin \theta }=a$
$\Rightarrow P=(a\cos \theta ,0)$
$Q=(0,a\sin \theta )$. Mid point of $PQ=(\frac{a\cos \theta }{2},\frac{a\sin \theta }{2})$
$\therefore$ locus is $4(x^2+y^2)=a^2$