First slide

Definition of a circle

Question

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

Moderate

A

a hyperbola
B

a parabola
C

an ellipse
D

a circle

Solution

Let $(h, k)$ be the centre of the circle. Since it touches $x -$ axis, its radius is $k$ . As it also touches the circle with centre $(0, 3)$ and radius $2$ (externally as $x -$ axis lies outside the given circle) distance between the centres is equals to the sum of their radii.
So $h^{2} + (k - 3)^{2} = (k + 2)^{2}$
$\Rightarrow h^{2} = 5 (2 k - 1)$
Locus of the centre $(h, k)$ is
$x^{2} = 5 (2 y - 1)$
which is a parabola.

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