Question

Let $O$ be the centre of the circle $x^2+y^2=r^2$, where $r>\frac{\sqrt{5}}{2}.$ 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 _____.

Answer 1

Answer: 2

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 PQ is $xh+yk=r^2$
Which is also $2x+4y=5$
$\Rightarrow (h,k)\equiv \left(\frac{2r^2}{5},\frac{4r^2}{5}\right)$
Mid-point of OR is circum-centre of $△OPQ$
$\Rightarrow \left(\frac{r^2}{5},\frac{2r^2}{5}\right)$ lies on $x+2y=4\Rightarrow r=2$