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Prove that the locus of the center of th...

Prove that the locus of the center of the circle which touches the given circle externally and the given line is a parabola.

A

a circle

B

an ellipse

C

a hyperbola

D

a pair of straight lines

Text Solution

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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)`.
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