In circular motion with constant speed, magnitude of instantaneous velocity, $v_{ins}=\omega r$.

Time taken to turn by $90°$ will be $=\frac{{\displaystyle \frac{\pi }{2}}}{\omega }=\frac{\pi }{2\omega }$.

Displacement will be equal to the length of  $AB=\sqrt{r^2+r^2}=r\sqrt{2}$

Average velocity

$$\begin{gathered} v_{avg}=\frac{\text{displacement}}{\text{time}} \\ =\frac{r\sqrt{2}\times 2\omega }{\pi }=\frac{2\sqrt{2}}{\pi }\omega r \end{gathered}$$

Hence, required ratio $\frac{v_{\text{ins }}}{v_{\text{avg }}}=\frac{\omega r\pi }{2\sqrt{2}\omega r}=\frac{\pi }{2\sqrt{2}}$

Therefore, $x=2$.