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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.
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