Question

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

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

Answer 1

The correct option is D a parabola

Equation of circle with centre (0, 3) and radius 2 is, $x^2+(y-3{)}^2=4$
Let locus of the variable circle is $(\alpha ,\beta )$ It touches x-axis.
It equation $(x-\alpha {)}^2+(y-\beta {)}^2={\beta }^2$
Given circles touch externally $C_1{C}_2=r_1+r_2$
$\Rightarrow \sqrt{{\alpha }^2+(\beta -3{)}^2}=2+\beta$
$\Rightarrow {\alpha }^2+(\beta -3{)}^2={\beta }^2+4+4\beta$
$\Rightarrow {\alpha }^2=10(\beta -1/2)\Rightarrow$ Locus is $x^2=10(y-1/2)$, which is parabola.