Question

The equation of a circle in parametric form is given by $x=a\cos \theta ,y=a\sin \theta$. The locus of the point of intersection of the tangents to the circle, whose parametric angles differ by $\frac{\pi }{2}$ is:

• A. $x^2+y^2=5a^2$
• B. $x^2+y^2=2a^2$
• C. $x^2+y^2=3a^2$
• D. $x^2+y^2=4a^2$

Answer 1

The correct option is B $x^2+y^2=2a^2$

The equation of the circle is $x^2+y^2=a^2$

Let $(a\cos \alpha ,a\sin \alpha )$ and $(a\cos (\alpha +\frac{\pi }{2}),a\sin (a+\frac{\pi }{2}))\equiv (-a\sin \alpha ,a\cos \alpha )$ be the points at which the tangents are drawn

Their equations are $x\cos \alpha +y\sin \alpha =a-\left(1\right)$ and $-x\sin \alpha +y\cos \alpha =a-\left(2\right)$

Squaring and adding the equations we get

$\Rightarrow x^2{\cos }^2\alpha +y^2{\sin }^2\alpha +x^2{\sin }^2\alpha +y^2{\cos }^2\alpha =a^2+a^2$

$\Rightarrow x^2({\cos }^2\alpha +{\sin }^2\alpha )+y^2({\sin }^2\alpha +{\cos }^2\alpha )=2a^2$

$\Rightarrow x^2+y^2=2a^2$

$\Rightarrow$The locus of point of intersection of tangents whole parametric angle is differ by $\frac{\pi }{2}$ is $x^2+y^2=2a^2$