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

Answer

A

Explanation

Let (h, k) be the coordinates of the centre of the circle. Since it touches x-axis. So, radius of the circle is |k|.
This circle also touches a circle of radius 2 having centre at (0, 3). Therefore, distance between their centre is equal to sum or difference of their radii.
i.e. `sqrt((h-0)^(2)+(k-3)^(2))=|k|pm2`
`rArr h^(2)+(k-3)^(2)=k^(2)+4 pm 4k`
`rArr h^(2)-10k+5=0 or, h^(2)-2k+5=0`
Hence, the locus of (h, k) is `x^(2)-10y+5=0 or x^(2)-2y+5=0` which are equations of a parabola.