MathNet 2021 — Part 03 Question 6

As shown in Fig. 11.1, in a plane rectangular coordinate system $xOy$, the left and right foci of 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 $Q_1(x_1, y_1), Q_2(x_2, y_2)$, respectively.

Find the maximum of $y_1 - y_2$.