Let O be the centre of the circle x2+y2=r2, where r>52. Suppose PQ is a chord of this circle and the equation of the line passing through P and Q is 2x+4y=5. If the centre of the circumcircle of the triangle OPQ lies on the line x+2y=4, then the value of r is _______

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Let $O$ be the centre of the circle $x^{2} + y^{2} = r^{2},$ where $r > \frac{\sqrt{5}}{2} .$ 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 the triangle $O P Q$ lies on the line $x + 2 y = 4,$ then the value of $r$ is ________

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answer is 2.

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Detailed Solution

Let $R (h, k)$ be point of intersection of tangents at $P$ and $Q$ on $x^{2} - y^{2} = r^{2}$
$\Rightarrow$ equation of chord of contact $P Q$ is $x h + y k = r^{2}$
which is also $2 x + 4 y = 5$
$\Rightarrow (h, k) \equiv (\frac{2 r^{2}}{5}, \frac{4 r^{2}}{5})$
Mid-point of $O R$ is circum-centre of $Δ O P Q$
$\Rightarrow (\frac{r^{2}}{5}, \frac{2 r^{2}}{5}) lies on x + 2 y = 4 \Rightarrow r = 2$