用动画生成这道题的解答过程---**Textual Information:** **Question Stem:** 已知双曲线 x²/a² - y²/b² = 1 (a>0, b>0) 的一条渐近线为 y=x, 且右焦点 F 到这条渐近线的距离为 √2. **Question (1):** 求双曲线的方程; **Question (2):** O为坐标原点, 过点 F 的直线 l 与双曲线的右支交于 A、B 两点, 与渐近线交于 C、D 两点, A 与 C 在 x 轴的上方, B 与 D 在 x 轴的下方. 设 S₁、S₂ 分别为 ΔAOC 的面积和 ΔBOD 的面积, 求 S₁+S₂ 的最大值. **Mathematical Formulas/Equations Identified:** * Hyperbola equation: x²/a² - y²/b² = 1 * Parameters condition: a > 0, b > 0 * Asymptote equation: y = x * Distance from right focus F to asymptote y=x is √2. **Other Relevant Text:** * O is the coordinate origin. * Line l passes through point F (right focus). * Points A, B are intersections of line l with the right branch of the hyperbola. * Points C, D are intersections of line l with the asymptotes. * A and C are above the x-axis. * B and D are below the x-axis. * S₁ is the area of triangle AOC. * S₂ is the area of triangle BOD. **Chart/Diagram Description:** * None. The image contains only text.