Question

If variable circle passes through the fixed point $(2,0)$ and touches $y$-axis, then the locus of center of circle is

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

Answer 1

The correct option is A a parabola.

Let centre of the circle be $P(h,k)$
Since circle touches $y$-axis $\Rightarrow$ radius of the circle is $h$
Thus equation of circle is given by,
$(x-h{)}^2+(y-k{)}^2=h^2$
Given it is also passing through $(2,0)$
$\Rightarrow (2-h{)}^2+k^2=h^2$
$\Rightarrow 4-4h+k^2=0\Rightarrow k^2=4(h-1)$
Hence required locus of $P(h,k)$ is given by, $y^2=4(x-1)$
which is clearly a parabola.