Prove that the locus of centre of circle which touches the given circle cxternally and given line is parabola.

Options

A. a circle
B. an ellipse
C. a hyperbola
D. a pair of straight lines

Explanation

Let `C_(1)` and `C_(2)` be the centres of two circles of radii `r_(1)` and `r_(2)` respectively. Let `P` be the centre and `r` be the radius of a circle touching the two given circles externally. Then,
`C_(1)P=r+r_(1)` and , `C_(2)P=r+r_(2)`
`impliesC_(1)P-C_(2)P=r_(1)-r_(2)=` Constant

Thus, point `P` moves in such a way that the difference of its distances from two fixed points `C_(1)` and `C_(2)` is constant. Hence, `P` describes a hyperbola whose two foci are at `C_(1)` and `C_(2)`.