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If a circle is drawn so as always to touch a given straight line and also a given circle externally then prove that the locus of its centre is a parabola.(given line and given circle are non intersecting)

If a circle is drawn so as always to touch a given straight line and also a given circle externally then prove that the locus of its centre is a parabola.(given line and given circle are non intersecting)

Grade:Upto college level

2 Answers

Chetan Mandayam Nayakar
312 Points
10 years ago

let O be centre of given circle, P be the centre of the circle whose locus is to be found, Q be the point of contact of the two circles, and PR be perpendicular to the given line.

PQ=PR, PQ+OQ=PR+OQ

thus the locus of P is such that its distance from a point O is equal to its perpendicular distance from a line( which is situated at a distance OQ from the given line, on the other side of the given circle). This description of the locus of P exactly fits the definition of a parabola.

Varun mishra
11 Points
4 years ago
Take any circle with centre O with centre r1 and P be the centre of circle (radius r) whose locus is to be found. Draw perpendicular to the given line say it is PM. Now produce PM to R such that PO is equal to PR.Now as PO is equal to PR , by classical definition of Parabola , Perpendicular Distance=Distance from fixed Pt.We may say that the locus of P is a Parabola

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