Question 4
As shown in Fig. 11.1, in a plane rectangular coordinate system $xOy$, the left and right foci of the ellipse $\Gamma : \frac{x^2}{2} + y^2 = 1$ are $F_1, F_2$, respectively. Let $P$ be a point on $\Gamma$ in the first quadrant, and the extensions of $PF_1, PF_2$ intersect $\Gamma$ at points $Q_1, Q_2$, respectively. Let $r_1, r_2$ be the radii of the incircles of $\triangle PF_1Q_2, \triangle PF_2Q_1$, respectively. Find the maximum of $r_1 - r_2$.