Question

数学这个怎么写

Answer 1

(1)
令P(x, y), OA与EP平行，则E(-1, y); EA的斜率为e = (y - 0)/(-1 -1) = -y/2
EA⊥FA, 则FA的斜率为-1/e = 2/y; 令F(-1, f), 2/y = (f - 0)/(-1 - 1) = -f/2, f = -4/y, F(-1, -4/y)
FO = (1, 4/y), OP = (x, y), 二者相互平行, 则x/1 = y/(4/y), 即y² = 4x

(2)
令直线的斜率为k, 其方程为y = kx + 2, 与抛物线联立并整理得k²x² + 4(k - 1)x + 4 = 0   (i)
要使其有解，需判别式16(k - 1)² - 16k² = 1 - 2k ≥ 0, k ≤ 1/2
后面的结果必须不与此矛盾。

令其二根的横坐标分别为m, n, 则m + n = -4(k - 1)/k², mn = 4/k²
M(m, km + 2), N(n, kn + 2）
AM = (m-1, km + 2), AN = (n - 1, kn + 2)
AM·AN = (m - 1)(n - 1) + (km + 2)(kn + 2) = (k + 1)mn + (2k - 1)(m + n) + 5
= 4(k+1)/k² -4(2k - 1)(k - 1)/k² + 5 = (16 - 3k)/k < 0
从16 - 3k和k异号得k < 0 (另一解k > 16/3与k ≤ 1/2矛盾，舍去)