Let C be the centre of the circle x ^ 2 + y ^ 2 - x + 2y = (11)/(4)...

Given,

${x}^{2} + {y}^{2} - x + 2y = \dfrac{11}{4}$

On rearranging terms we get,

$⇒ {\left(x - \dfrac{1}{2}\right)}^{2} + {\left(y + 1\right)}^{2} = {\left(2\right)}^{2}$

Now given in $∆ PQR$, $∠ PCR = \dfrac{\pi}{4}$ so $∠ PQR = 22\dfrac{1}{2}$ by circle property, radius is $2$ so $RC = QC = 2$

Now, $PR = QRsin22\dfrac{1}{2}$$= 4sin\dfrac{\pi}{8}$

And $PQ = QRcos22\dfrac{1}{2} = 4\cos 22\dfrac{1}{2}$

Now area of $\delta PQR = \dfrac{1}{2}PR \times PQ$

$= \dfrac{1}{2}\left(4sin\dfrac{\pi}{8}\right)\left(4cos\dfrac{\pi}{8}\right)$

$= 4 \times \left(2sin\dfrac{\pi}{8}cos\dfrac{\pi}{8}\right)$

$= 4sin\dfrac{\pi}{4} = \dfrac{4}{\sqrt{2}} = 2\sqrt{2}$