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Let $O$ be the centre of the circle $x^{2} + y^{2} = r^{2},$ where $r > \sqrt{\frac{5}{2}} .$ Suppose, $P Q$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2 x + 4 y = 5 .$ If the centre of the circumcircle of $Δ O P Q$ lies on the line $x + 2 y = 9,$ then the value of $r$ is

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Detailed Solution

$O A = \frac{\sqrt{5}}{2}, O C = \frac{9}{\sqrt{5}}$

$C Q = O C = \frac{9}{\sqrt{5}}$ and $C A = \frac{13}{2 \sqrt{5}}$

$∴ O Q = \sqrt{O A^{2} + A Q^{2}} = \sqrt{O A^{2} + {(C Q^{2} - C A)}^{2}} = \sqrt{\frac{5}{4} + \frac{81}{5} - \frac{169}{20}} = \sqrt{9} = 3 .$

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