A square $OABC$ is formed by line pairs $xy = 0$ and $xy + 1 = x + y$ where ' $O$ 'is the origin. A

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A square $OABC$ is formed by line pairs $xy = 0$ and $xy + 1 = x + y$ where ' $O$ 'is the origin. A circle with centre $C_{1}$ inside the square is drawn to touch the line pair $xy = 0$ and another circle with centre $C_{2}$ and radius twice that of $C_{1}$, is drawn to touch the circle $C_{1}$ and the other line pair. The radius of the circle with centre $C_{1}$ is:

Accepted Answer

[Key Concept: From the line-pairs $xy = 0$ and $xy + 1 = x + y$ find the square's vertices, then read the property.]$C_{1},C_{2}$ lie on $y = x$ line$$\begin{matrix} OB = OC_{1} + C_{1}C_{2} + \ \sqrt{2} = r\sqrt{2} + 3r + 2r \ \ r = \frac{\sqrt{2}}{3(\sqrt{2} + 1)} \end{matrix}$$

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