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
$(0-1)^{2}+(y-1)^{2}=(1+|y|)^{2}$
$\Rightarrow 1+y^{2}+1-2 y=1+y^{2}+2|y|$
If $y>\,0$, $y^{2}+2-2 y=y^{2}+1+2 y$
$\Rightarrow 4 y=1$
$\Rightarrow y=\frac{1}{4}$
If $y<\,0$, $y^{2}+2-2 y=y^{2}+1-2 y$
$\Rightarrow 1=2$ (Not possible)
$\therefore \quad y=\frac{1}{4}$