A circle touches the X-axis and also touches another circle with centre at (0, 3) and radius 2. Then the locus of the centre of the first circle is
Let the centre be \((h, k)\) with \(k > 0\). It touches the x-axis, so its radius is \(k\).
It touches the circle with centre (0, 3) and radius 2 externally, so the distance between centres = \(k + 2\):
\(h^2 + (k-3)^2 = (k+2)^2 \Rightarrow h^2 - 6k + 9 = 4k + 4 \Rightarrow h^2 = 10k - 5\)
Locus: \(x^2 = 10y - 5\), which is a parabola.
Answer: (A)
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