A circle touches the x-axis and also touches the circle with centre at (0,3) and radius 2.

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:

Accepted Answer

Equation of circle with center  $(0,3)$ and radius $2$ is

$${x^2} + {\left( {y - 3} \right)^2} = 4$$

Let locus of the variable circle is $$\left( {\alpha ,\beta } \right)$$

As it  touches $x$-axis.

$$\therefore$$ It's equation is $${\left( {x - \alpha } \right)^2} + {\left( {y + \beta } \right)^2} = {\beta ^2}$$

Circle touch externally $$ \Rightarrow {c_1}{c_2} = {r_1} + {r_2}$$

$$\therefore$$ $$\sqrt {{\alpha ^2} + {{\left( {\beta  - 3} \right)}^2}}  = 2 + \beta $$

$${\alpha ^2} + {\left( {\beta  - 3} \right)^2} = {\beta ^2} + 4 + 4\beta $$

$$ \Rightarrow {\alpha ^2} = 10\left( {\beta  - 1/2} \right)$$

$$\therefore$$ Locus is $${x^2} = 10\left( {y - {1 \over 2}} \right)$$ which is parabola.

Solutions · 1