A geometry problem by Sri Chakra

Nice problem, Sri Chakra! (although I think it should be $\frac{r^4}{2|a||b|}$ . Anyway, only considering positive a, b, I did it as follows:

diagram

By a small angle chase we can see that $OXA$ ~ $OPQ$ and that $OA$ , the radius $r$ say, is in proportion to $OPQ$ as the red line is to $OXA$ .

Now by considering areas we deduce that the red line is of length $\frac{ab}{r}$ and so the linear scale factor from $OXA$ to $OPQ$ is $\frac{r^2}{ab}$ . Hence the area scale factor is $\frac{r^4}{(ab)^2}$ .

The area of OXA is $\frac{ab}{2}$ so the area of OPQ is $\frac{r^4}{2ab}$ as required.