Question

The locus of the point of intersection of the tangents to the circle $x=r\cos \theta ,y=r\sin \theta$ at points whose parametric angels differ by $\frac{\pi }{3}$

• A. $x^2+y^2=4(2-\sqrt{3})r^2$
• B. $3(x^2+y^2)=r^2$
• C. $x^2+y^2=(2-\sqrt{3})r^2$
• D. $3(x^2+y^2)=4r^2$

Answer 1

Answer: (D)

$3(x^2+y^2)=4r^2$

Tangent at $\theta =0$ and $\theta =\frac{\pi }{3}$ are $x=r$
and $x\cos \left(\frac{\pi }{3}\right)+y\sin \left(\frac{\pi }{3}\right)=r$
$\Rightarrow x=r,y=\frac{r}{\sqrt{3}}\Rightarrow 3(x^2+y^2)=4r^2$