A circle with center (2,2) touches the coordinate axes and a straight line AB where A and B lie on positive direction of coordinate axes such that the circle lies between origin and the line AB. If O be the origin then the locus of circumcenter of $Δ O A B$ will be:

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$x y = x + y - \sqrt{x^{2} + y^{2}}$

$x y = x + y + \sqrt{x^{2} + y^{2}}$

$x y + x + y = \sqrt{x^{2} + y^{2}}$

$x y + x + y + \sqrt{x^{2} + y^{2}} = 0$

answer is A.

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Detailed Solution

Circum center is mid point of AB = $(x_{1}, y_{1})$
Equation of AB is $\frac{x}{2 x_{1}} + \frac{y}{2 y_{1}} = 1 \Rightarrow y_{1} x + x_{1} y = 2 x_{1} y_{1}$

$d = r \Rightarrow \frac{|y_{1} 2 + x_{1} 2 - 2 x_{1} y_{1}|}{\sqrt{x_{1}^{2} + y_{1}^{2}}} = 2 \Rightarrow \frac{- (y_{1} 2 + x_{1} 2 - 2 x_{1} y_{1})}{\sqrt{x_{1}^{2} + y_{1}^{2}}} = 2 L o c u s i s x y = x + y + \sqrt{x^{2} + y^{2}}$