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

  an ellipse
• B

  a circle
• C

  a hyperbola
• Correct answer
  D

  a parabola

Solution

Equation of circle with centre $(0,3)$ and radius 2 is $x^2+(y-3)^2=4$ Let locus of the variable circle is $(\alpha, \beta)$ $\because$ It touches $x$-axis. $\therefore$ It equation $(x-\alpha)^2+(y-\beta)^2=\beta^2$ Circles touch externally $$ \begin{aligned} & \therefore \sqrt{\alpha^2+(\beta-3)^2}=2+\beta \\ & \alpha^2+(\beta-3)^2=\beta^2+4+4 \beta \\ & \alpha^2=10(\beta-1 / 2) \end{aligned} $$ $\therefore$ Locus is $x^2=10(y-1 / 2)$ which is parabola.

Hence, the correct answer is option: 'D'.