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

Explanation

Let the given circle `x^(2)+y^(2)=a^(2)` and given line be x=b.
One of the variable circles C having centre P(h,k) touches the given circle at A and line at B.
From the figure, radius of circle C is (b-h).
Also, OP=OA+AP
`:." "OP=OA+PB`
`rArr" "sqrt(h^(2)+k^(2))=a+(b-h)`
`rArr" "(a+b)^(2)-2(a+b)h+h^(2)=h^(2)+k^(2)`
`rArr" "y^(2)=(a+b)^(2)-2(a+b)x`,
which is required equation of locus.
Clearly, this is equation of parabola.