Question

Two circles are given such that one is completely lying inside the other without touching. Then the locus of the centre of a variable circle which touches the smaller circle from outside and the bigger circle from inside is

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

Answer 1

Answer: (A)

In the figure, circles with hard lines are the given circles with centres $C_{1}$ and $C_{2}$ and radii $r_{1}$ and $r_{2}$.
Let the circle with dotted line be the variable circle, which touches the given two circles as explained in the question, which has centre $C$ and radius $r$.
Now $C C_{2}=r+r_{2}$ and $C C_{1}=r_{1}-r$
Hence, $C C_{1}+C C_{2}=r_{1}+r_{2}$ (=constant)
Then the locus of $C$ is an ellipse whose foci are $C_{1}$ and $C_{2}$.