Question

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

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

Answer 1

Answer: (B)

a parabola

Let locus of centre be $\alpha ,\beta$ then according to given, if $r_1,r_2$,are radii of circles then $C_1{C}_2=r_2\pm r_1$
$\Rightarrow (\alpha -0{)}^2+(\beta -3{)}^2=|\beta \pm 2|$
$\Rightarrow {\alpha }^2+{\beta }^2-6\beta +9={\beta }^2+4+4\beta$
and ${\alpha }^2+{\beta }^2-6\beta +9={\beta }^2-4\beta +4$
$\Rightarrow {\alpha }^2-10\beta +5=0$ and $x^2-2\beta +5$
$\Rightarrow x^2-10y-5$ and $x^2=2y-5$ both are parabolas but $x^2=2y-5$ does not exist
Ans: B