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 $2 x +4 y =5$. If the centre of the circumcircle of the triangle OPQ lies on the line $x+2 y=4$, then the value of $r$ is _____.

Answer 1

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