Question

Let C be the circle with center 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. $\frac{1}{4}$
• B. $\frac{\sqrt{3}}{\sqrt{2}}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $\frac{1}{4}$

Since circle T passes through origin and its centre is on the y-axis $(0,y)$
Its radius will be $y$
Now given both circle touches each other externally,
$\Rightarrow CT=r_1+r_2$ $\Rightarrow \sqrt{(1+(1-y{)}^2}=1+y$
$\Rightarrow 1+1+y^2-2y=1+y^2+2y\Rightarrow y=\frac{1}{4}$, which is radius of the circle T.