Given,

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

On rearranging terms we get,

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

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

Now, $PR=QR\sin 22\frac{1}{2}$$=4\sin \frac{\pi }{8}$

And $PQ=QR\cos 22\frac{1}{2}=4\cos 22\frac{1}{2}$

Now area of $\Delta PQR=\frac{1}{2}PR\times PQ$

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

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

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