Two circles are given such that they neither intersect nor touch. Then identify the locus of the center of variable circle which touches both the circles externally.

Options

A. a circle
B. an ellipse
C. a hyperbola
D. none of these

Answer

C

Explanation

Let `C_(1), C_(2)` be the centres of two given circles of radii `r_(1) and r_(2)` respectively. Let C be the centre of a circle of radius r which touches the two circles externally. Then,
`C C_(1)=r+r_(1) and C C_(2)=r+r_(2)`
`rArr C C_(1)-C C_(2)=r_(1)-r_(2)`
`rArr ` c lies on a hyperbola having two foci at `C_(1) and C_(2)`.
Hence, the locus of C is a hyperbola.