Question

From a point $P$ on the circle $C_1:x^2+y^2=25$, tangents are drawn to the circle $C_2:x^2+y^2=9$. Let $\theta$ be the angle made by the line $PO$ ($O$ is the origin) with positive $x$-axis. If the chord of contact of the circle $C_2$ is tangent to another circle $C_3$ which passes through the origin, then locus of center of the circle $C_3$ is

• A. a parabola with focus at $(0,0)$ and for $\theta =\frac{\pi }{4}$, vertex is $(\frac{9\sqrt{2}}{20},\frac{9\sqrt{2}}{20})$
• B. a parabola with focus at $(0,0)$ and for $\theta =0$, vertex is $(0,\frac{9}{10})$
• C. a parabola with focus at $(0,3\sin \theta )$ and for $\theta =0$, vertex is $(\frac{9\sqrt{3}}{20},0)$
• D. a parabola with focus at $(0,0)$ and for $\theta =\frac{\pi }{4}$, vertex is $(\frac{9}{10},0)$

Answer 1

The correct option is A a parabola with focus at $(0,0)$ and for $\theta =\frac{\pi }{4}$, vertex is $(\frac{9\sqrt{2}}{20},\frac{9\sqrt{2}}{20})$

Any point on the circle $C_1$ is $P(5\cos \theta ,5\sin \theta )$
Chord of contact of the tangents drawn from the $P$ is
$5x\cos \theta +5y\sin \theta =9\Rightarrow x\cos \theta +y\sin \theta =\frac{9}{5}$

Let the center of $C_3$ is $(h,k)$
If the radius of $C_3$ is $r$,
then $\sqrt{h^2+k^2}=r$
and $\frac{|5h\cos \theta +5k\sin \theta -9|}{5}=r$
$\therefore$ locus of the point $(h,k)$ is a parabola whose focus is at the origin and $x\cos \theta +y\sin \theta =\frac{9}{5}$ as directrix.

For $\theta =\frac{\pi }{4},h=k\Rightarrow \frac{|5h+5k-9\sqrt{2}|}{5\sqrt{2}}=\sqrt{h^2+k^2}\therefore 5h+5k-9\sqrt{2}=-5\sqrt{2}\left(\sqrt{h^2+k^2}\right)(\because \text{centre and origin lie on the same side in the}{1}^{st}\text{quadrant})$
$(h,k)\equiv (\frac{9\sqrt{2}}{20},\frac{9\sqrt{2}}{20})$