Question

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 $C$.

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

Answer 1

Answer: (B)

An ellipse

Answer 2

Let $O_1, R_1$ be the center and radius of $C_1$; $O_2, R_2$ be the center and radius of $C_2$; and $O, r$ be the center and radius of $C$. Since $C$ touches $C_1$ internally, $d(O_1, O) = R_1 - r$. Since $C$ touches $C_2$ externally, $d(O_2, O) = R_2 + r$. Adding these two equations, we get $d(O_1, O) + d(O_2, O) = (R_1 - r) + (R_2 + r) = R_1 + R_2$. This means the sum of the distances from $O$ to two fixed points ($O_1$ and $O_2$) is a constant ($R_1 + R_2$). This is the definition of an ellipse where $O_1$ and $O_2$ are the foci. The condition that $C_2$ lies inside $C_1$ implies $d(O_1, O_2) < R_1 - R_2$. This ensures that the sum of the distances to the foci ($R_1 + R_2$) is greater than the distance between the foci ($d(O_1, O_2)$), confirming the locus is an ellipse.