JEE Advanced 2020 — Paper 2 — Question 36

Let $R(h, k)$ be point of intersection of tangents at $P$ and $Q$ on $x^{2}-y^{2}=r^{2}$

$\Rightarrow \quad$ equation of chord of contact PQ is $\mathrm{xh}+\mathrm{yk}=\mathrm{r}^{2}$

Which is also $2 x+4 y=5$

$\Rightarrow \quad(\mathrm{h}, \mathrm{k}) \equiv\left(\frac{2 \mathrm{r}^{2}}{5}, \frac{4 \mathrm{r}^{2}}{5}\right)$

Mid-point of OR is circum-centre of $\triangle \mathrm{OPQ}$

$\Rightarrow \quad\left(\frac{r^{2}}{5}, \frac{2 r^{2}}{5}\right)$ lies on $x+2 y=4 \Rightarrow r=2$