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:

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

Answer 1

Answer: (A)

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
$x^2=-5+6y+4|y|$
$x^2=10y-5\left(\because y>0\right)$
This equation represents a parabola. Thus locus of the centre of the circle is a parabola.