Question

Let $C$ be the circle with centre at $(1,1)$ and radius $1$. If $T$ is the circle centred at $(0,y)$ passing through origin and touching the circle $C$ externally, then the radius of $T$is equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\frac{\sqrt{3}}{\sqrt{2}}$

Answer 1

Answer: (B)

$C\equiv (x-1{)}^2+(y-1{)}^2=1$
Radius of $T=|y|$
$T$ touches $C$ externally
$(0-1{)}^2+(y-1{)}^2=(1+|y|{)}^2$
$\Rightarrow 1+y^2+1-2y=1+y^2+2|y|$
If $y>0$, $y^2+2-2y=y^2+1+2y$
$\Rightarrow 4y=1$
$\Rightarrow y=\frac{1}{4}$
If $y<0$, $y^2+2-2y=y^2+1-2y$
$\Rightarrow 1=2$ (Not possible)
$\therefore y=\frac{1}{4}$