The locus of the centre of a circle which pas | vedclass

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(C) Let the centre of the circle be $(h, k)$.Since the circle passes through the point $(a, 0)$,the radius $r$ is the distance between $(h, k)$ and $(

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Parabola

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(C) Let the centre of the circle be $(h, k)$.Since the circle passes through the point $(a, 0)$,the radius $r$ is the distance between $(h, k)$ and $(a, 0)$,so $r = \sqrt{(h - a)^2 + k^2}$.The circle also touches the line $x + 1 = 0$,so the radius $r$ is the perpendicular distance from $(h, k)$ to the line $x + 1 = 0$,which is $r = |h + 1|$.Equating the two expressions for $r^2$,we get $(h - a)^2 + k^2 = (h + 1)^2$.Expanding this,$h^2 - 2ah + a^2 + k^2 = h^2 + 2h + 1$.Simplifying,$k^2 = 2h(1 + a) + 1 - a^2$.Replacing $(h, k)$ with $(x, y)$,the locus is $y^2 = 2x(1 + a) + (1 - a^2)$,which represents a parabola.

The locus of the centre of a circle which passes through the point $(a, 0)$ and touches the line $x + 1 = 0$ is

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