Question

Let $C$ be the circle with centre $(0,0)$ and radius $3$ units. Locus of the point $P$ from which the chord of contact subtends an angle of ${60}^{\circ }$ at any point on the circumference of $C$ is

• A. $x^2+y^2=27$
• B. $x^2+y^2=18$
• C. $x^2+y^2=36$
• D. $x^2+y^2=16$

Answer 1

The correct option is C $x^2+y^2=36$

$C:x^2+y^2=9$

In triangle $OAB$,
$AB=\left(2\sin {60}^{\circ }\right)OA$

$\Rightarrow \frac{2\left(\sqrt{h^2+k^2-9}\right)\cdot 3}{\sqrt{h^2+k^2}}=\frac{3\cdot 2\cdot \sqrt{3}}{2}\left[\text{From}△OAP\right]$
$\Rightarrow 4(h^2+k^2-9)=3(h^2+k^2)$
$\Rightarrow h^2+k^2=36$

Hence, locus is $x^2+y^2=36$.