Question

Let $S_1$ and $S_2$ be two circles with $S_2$ lying inside $S_1$. A circle $S$ lying inside $S_1$ touches $S_1$ internally and $S_2$ externally. The locus of the center of $S$ is a/an

• A. parabola
• B. ellipse
• C. hyperbola
• D. circle

Answer 1

The correct option is B ellipse

We choose centre of $S_1$ as the origin and the line joining the centres of $S_1,S_2$ as the X-axis .

Let the centre of $S_2$ be $A(a,0)$ and $b,c(b>c)$ be the radii if $S_1,S_2,$ respectively.

If $P(h,k)$ be the centre of the variable circle $S$ and $r$ be its radius then we have $OP+r=b$

i.e., $\sqrt{h^2-k^2}=b-r$ ...(1)

and, $AP=r+c$

i.e., $\sqrt{{(h-a)}^2+k^2}=r+c$ ...(2)

Eliminating $R$ from equations (1) and (2), we have

$OP+AP=b+c$

i.e., sum of distances of $P$ from two fixed points $O$ and $A=$

constant.

Hence, $P$ lies on an ellipse having foci at $O$ and $A.$