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If the Circumcenter of the triangle formed by the points $A(a, 3), B(b, 5)$ and $C(a, b)$ is $(1,1)$, then out of all the possible coordinates of $\mathrm{C}$ the sum of the absolute values of the distinct coordinates of $\mathrm{C}$ is
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9
$\because \mathrm{OP} \perp \mathrm{AC}$
$\Rightarrow 12=(\mathrm{a}-\mathrm{b})(\mathrm{a}+\mathrm{b}-2)$ ...(i)
Putting the value of $b=-1$ and $b=3$ in equation (i) we get coordinate of $\mathrm{C}$ are $(5,3),(-3,3)(5,-1)$ and $(-3,-1)$ Now sum of the absolutes values of the distinct coordinates of $\mathrm{C}$ is $5+|-3|+|-1|=9$.
$(\because(5,3)$ and $(-3,3)$ are same with $A)$.
$\Rightarrow 12=(\mathrm{a}-\mathrm{b})(\mathrm{a}+\mathrm{b}-2)$ ...(i)
Putting the value of $b=-1$ and $b=3$ in equation (i) we get coordinate of $\mathrm{C}$ are $(5,3),(-3,3)(5,-1)$ and $(-3,-1)$ Now sum of the absolutes values of the distinct coordinates of $\mathrm{C}$ is $5+|-3|+|-1|=9$.
$(\because(5,3)$ and $(-3,3)$ are same with $A)$.
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