Question

高中数学，第11题，要过程

Answer 1

F(c, 0), 过一三象限的渐近线为y = bx/a, 其斜率为b/a, 在△OMP中, OP为直径，则∠OMP为直角, FP的斜率为-a/b, 方程为y = (-a/b)(x - c)与上述渐近线联立, 得到M的坐标为(a²/c, ab/c)
另外, 令y = (-a/b)(x - c)中的x = 0, y = ac/b, P(0, ac/b)
△MFO的面积 = (1/2)*|OF|*M的纵坐标 = (1/2)c*ab/c = ab/2
△PMO的面积 = (1/2)*|OP|*M的横坐标 = (1/2)(ac/b)*(a²/c) = a³/(2b)
按题意: ab/2 = 5a³/(2b), b² = 5a²
e² = c²/a² = (a² + b²)/a² = (a² + 5a²)/a² = 6
答案C