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. $\left\{\right(x,y):x^2=4y\}$
• B. $\left\{\right(x,y):x^2+(y-1{)}^2=4\}$
• C. $\left\{\right(x,y):x^2=y\}$
• D. $\left\{\right(x,y):x^2=4y\}\cup \left\{\right(0,y):y<0\}$

Answer 1

Answer: (D)

$\left\{\right(x,y):x^2=4y\}\cup \left\{\right(0,y):y<0\}$

All the circles having center on the negative y-axis and passing though the origin touch the circle $x^2+{(y-1)}^2=1$ and the x-axis

If $(x,y)$ is center of the circle $C$, then

$r_1+r_2=A_1{A}_2\Rightarrow 1+\left|y\right|=\sqrt{x^2+{(y-1)}^2}\Rightarrow 1+y^2+2\left|y\right|=x^2+y^2-2y+1$

$\Rightarrow x^2+2\left|y\right|+2y=0\Rightarrow x^2=4y$ if $y\ge 0$

Thus locus of the center is

$\{(x,y)|x^2=4y\}\cup \{(0,y)|y\le 0\}$