IIT-JEE 2001Question 1
Let $C_1$ and $C_2$ be two circles with $C_2$ lying inside $C_1$. A circle $C$ lying inside $C_1$ touches $C_1$ internally and $C_2$ externally. Identify the locus of the centre of $C$.
Circle
AnswerAn ellipse (foci at the centres of $C_1$ and $C_2$)
Solution
Concept. Internal contact subtracts radii, external contact adds them. The radius $r$ of $C$ cancels in the sum. Formulas used. (1) Internal contact: $d=R-r$. External contact: $d=R+r$.(2) $PS_1+PS_2=2a>S_1S_2$: ellipse with foci $S_1, S_2$.Step 1 - Notation. $C_1$: centre $O_1$, radius $R_1$. $C_2$: centre $O_2$, radius $R_2$. $C$: centre $P$, radius $r$. Step 2 - Use (1). $PO_1=R_1-r$ and $PO_2=R_2+r$. Step 3 - Add. $PO_1+PO_2=R_1+R_2$, a constant.Step 4 - Use (2). $C_2$ is inside $C_1$, so $O_1O_2<R_1-R_2<R_1+R_2$. The locus is an ellipse with foci $O_1, O_2$ and major axis $R_1+R_2$. Answer: An ellipse (foci at the centres of $C_1$ and $C_2$)