Question

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

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

Answer 1

Answer: (D)

$\frac{1}{4}$

Distance between there centers,

$\Rightarrow$ $k+1=\sqrt{1+(k-1{)}^2}$

$\Rightarrow$ $k+1=\sqrt{1+k^2+1-2k}$

$\Rightarrow$ $k+1=\sqrt{k^2+2-2k}$

$\Rightarrow$ $(k+1{)}^2=k^2+2-2k$

$\Rightarrow$ $k^2+2k+1=k^2+2-2k$

$\Rightarrow$ $4k=1$

$\therefore$ $k=\frac{1}{4}$