(1) 显然P只能在y = 5/4下方
P与F(0,-1/4)的距离比其与y = 5/4的距离小1,相当于P与F(0,-1/4)的距离等于与y = 1/4的距离
P的轨迹为以y = 1/4为准线,F(0,-1/4)为焦点的抛物线,p/2 = 1/4,p = 1/2,方程x² = -y
(2)
x² = -y = -mx + 4
x² + mx - 4 = 0
x₁ + x₂ = -m,x₁x₂ = -4
|AB|² = (x₁ - x₂)² + (y₁ - y₂)² = (x₁ - x₂)² + (mx₁ - 4 - mx₂ + 4)² = (m² + 1)(x₁ - x₂)²
= (m² + 1)[(x₁+ x₂)² - 4x₁x₂]
= (m² + 1)(m² + 16) = 54
(m² + 19)(m² - 2) = 0
m = ±√2