Q) A circle touches the x axis and also touches another fixed circle with center (0,3) and radius = 2 units. What is the locus of the center of the (first) circle?

A) Consider any such arbitrary circle, with center (h,k).
It touches the x axis, so its radius is k units ( the ordinate).

Now, the distance of this circle's center from the given circle's center = 2 + k units
This distance is the same as the distance between center of the arbitrary circle and the line y = -2 .

Now, since the center of this arbitrary circle is equidistant from a fixed point (the center of given circle ie. (0,3) ) and the line y = -2, it follows the definition of the parabola and hence the locus of (h,k) is a parabola.

So the locus of the center of the circle is a parabola with focus (0,3) and directrix y = -2 .

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