Question

lf a variable circle $C$ touches the $x$-axis and touches the circle $x^2+(y-1{)}^2=1$
externally, then the locus of centre of $C$ can be

• A. $x^2=4y$
• B. $(x-1{)}^2+y^2=1$
• C. $x=4y^2$
• D. $x=0andy<0$

Answer 1

Answer: (A), (D)

The correct options are
A $x^2=4y$
D $x=0andy<0$

Let $e{q}^n$ of circle touching x-axis is
$x^2+y^2+2gx+2f+g^2=0$
and this circle touche $x^2+(y-1{)}^2=1$ externally so,
$\Rightarrow c_1{c}_2=r_1+r_2$
$\Rightarrow \sqrt{g^2+(1+f{)}^2}=1+\sqrt{g^2+f^2-g^2}$

If f is negative

$g^2+(1+f{)}^2=(1-f{)}^2$
$g^2=-4f$
So, locus of centre $(-g,-f)$ is
$-g\to x$ and $-f\to y$
$\Rightarrow x^2=4y$

If f is positive

$g^2+(1+f{)}^2=(1+f{)}^2$
$g=0$
So, locus of centre $(-g,-f)$ is
$-g\to x$
$\Rightarrow x=0$