Question 76
Given an ellipse equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a \gt b \gt 0$) in a plane rectangular coordinate system $xOy$, let $A_1, A_2, F_1, F_2$ be its left and right end-points, left and right focuses, respectively, and $P$ be any point on the ellipse different from $A_1, A_2$. Suppose there are points $Q, R$ satisfying $QA_1 \perp PA_1, QA_2 \perp PA_2, RF_1 \perp PF_1, RF_2 \perp PF_2$. Find and prove the relationship between the length of segment $QR$ and $b$.