Question

A circle is given by $x^2+(y-1{)}^2=1$, another circle $C$ touches it externally and also the $x$-axis, then the locus of its centre is

• A. $x^2=2y$
• B. $x^2=3y$
• C. $x^2=y$
• D. $x^2=4y$

Answer 1

The correct option is D $x^2=4y$

$x^2+(y-1{)}^2=1$
Centre is $A(0,1)$
Let the centre of circle $C$ be $(h,k)$.
Since, the circle touches $x$-axis and the given circle $x^2+(y-1{)}^2=1$ externally, so its radius $=k$


Also, the distance between centres = sum of radii
$\Rightarrow \sqrt{(h-0{)}^2+(k-1{)}^2}=1+k$
$\Rightarrow h^2+k^2-2k+1=1+2k+k^2$
$\Rightarrow h^2=4k$

$\therefore$ Locus of $(h,k)$ is $x^2=4y$