A circle touches the `x`-axis and also touches the circle with center `(0, 3)` and radius `2`. The locus of the center

Options

A. a circle
B. an ellipse
C. a parabola
D. a hyperbola

Explanation

(3) Let `C_(1)(h,k)` be the centre of the circle.
The circle touches the c-axis. Then its radius is `r_(1)=k`
Also, the circle touches the circle with center `C_(2)(0,3)` and radius `r_(2)=2`. Therefore,
`|C_(1)C_(2)|=r_(1)+r_(2)`
`orsqrt((h-0)^(2)+(k-3)^(2))=|k+2|`
Squaring, we get
`h^(2)-10k+5=0`
Therefore, the locus is `x^(2)-10y+5=0`, which is a parabola.