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

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

Tardigrade Admin · Accepted Answer

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}

.

Q. 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

An ellipse

A hyperbola

A parabola

A circle