Q. A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is:
AIEEEAIEEE 2005Conic Sections
Solution:
Since circle touches the x-axis and also touches circle with the centre at $(0, 3)$ and radius $2$, then
$C_{1}C_{2} = r_{1} + r_{2}$
$h^{2}+\left(k-3\right)^{2}= \left(\left|k\right|+2\right)^{2}$
$h^{2}+k^{2}+9-6k = k^{2} + 4\left|k\right|$
$\therefore $ Locus of centre of circle is
$x^{2} = - 5+ 6y + 4|y|$
$x^{2} = 10y - 5\quad\left(\because y > 0\right)$
This equation represents a parabola. Thus locus of the centre of the circle is a parabola.
