Question

The locus of center of the circle which touches the circle $x^2+{(y-1)}^2=1$ externally and also touches x-axis is

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

Answer 1

The correct option is B $\{(x,y):x^2=4y:y>0\}\cup \{(0,y):y<0\}$

According to given condition

$\sqrt{{(h-0)}^2+{(k-1)}^2}=1+\left|k\right|\Rightarrow h^2+{(k-1)}^2={(1+\left|k\right|)}^2\Rightarrow h^2=2k+2\left|k\right|$

Hence locus is $x^2=2y+2\left|y\right|$

Clearly for $y>0,x^2=4y$

and for $y<0,x^2=0\Rightarrow x=0$