Testing linear-quadratic GUP modified Kerr Black hole using EHT results

The linear-quadratic Generalized uncertainty principle (LQG) is consistent with predictions of a minimum measurable length and a maximum measurable momentum put forth by various theories of quantum gravity. The quantum gravity effect is incorporated into a black hole (BH) by modifying its ADM mass. In this article, we explore the impact of GUP on the optical properties of an LQG modified \k BH (LQKBH). We analyze the horizon structure of the BH, which reveals a critical spin value of $7M/8$. BHs with spin $(a)$ less than the critical value are possible for any real GUP parameter $\a$ value. However, as the spin increases beyond the critical value, a forbidden region in $\a$ values pops up that disallows the existence of BHs. This forbidden region widens as we increase the spin. We then examine the impact of $\a$ on the shape and size of the BH shadow for inclination angles $17^o$ and $90^o$, providing a deeper insight into the unified effect of spin and GUP on the shadow. The size of the shadow has a minimum at $\a=1.0M$, whereas, for the exact value of $\a$, the deviation of the shadow from circularity becomes maximum when the spin is less than the critical value. No extrema is observed for $a\,>\, 7M/8$. The shadow's size and deviation are adversely affected by a decrease in the inclination angle. Finally, we confront theoretical predictions with observational results for supermassive BHs $M87^*$ and $SgrA^*$ provided by the EHT collaboration to extract bounds on the spin $a$ and GUP parameter $\a$. We explore bounds on the angular diameter $\th_d$, axial ratio $D_x$, and the deviation from \s radius $\d$ for constructing constraints on $a$ and $\a$. Our work makes LQKBHs plausible candidates for astrophysical BHs.