Question

let C be with the circle with the centre at $\left(1,1\right)$ and radius 1, if the circle T is the circle centred at $\left(0,y\right)$, passing through origin and touching the circle centrally, the radius of T is

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

Answer 1

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

We have,

The centre of circle $C=(1,1)$

Radius $r=1$

$T$ is the circle centred at point $(0,y)$

Then, we know that

$CT=1+y$

So, using distance formula,

$CT=\sqrt{{(1-0)}^2+{(1-y)}^2}$

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

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

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

$\Rightarrow 4y=1$

$\Rightarrow y=\frac{1}{4}$

Hence, this is the answer.