Question

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

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

Answer 1

The correct option is C a parabola

Let $C_1(h,k)$ be the center of the circle.

Circle touches the $x$-axis then its radius is $r_1=k$.

Also circle touches the circle with centre $C_2(0,3)$ and radius $r_2=2$.

$\therefore |C_1{C}_2|=r_1+r_2$

$\Rightarrow \sqrt{(h-0{)}^2+(k-3{)}^2}=|k+2|$

Squaring

$h^2-10k+5=0$

$\Rightarrow$ Locus is $x^2-10y+5=0$, which is parabola.