Question

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

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

Answer 1

Answer: (A)

Let the centre of a circle be $C_{1}(h, k)$
Since, $C_{1} C_{2}=r_{1} +r_{2}$ (given)
$\Rightarrow \sqrt{(h-0)^{2}+(k-3)^{2}}=|k+2|$
$\Rightarrow h^{2}=5(2 k-1)$
Hence, locus of a point is $x^{2}=5(2 y-1)$ which represents a equation of parabola.