Question

过抛物线y²=2px(p＞0)的焦点F,且倾斜角为π/4的直线与抛物线交于A,B两点，若弦AB的垂直

过抛物线y²=2px(p＞0)的焦点F,且倾斜角为π/4的直线与抛物线交于A,B两点，若弦AB的垂直平分线经过(0,2),则p等于

Answer 1

焦点为F(p/2, 0), 斜率为k = tan(π/4) = 1, AB：y = x - p/2, x = y + p/2
代入抛物线：y² = 2p(y + p/2) = 2py + p²
y² - 2py - p² = 0
y1 + y2 = 2p, y1*y2 = -p²
AB的中点为M(m, m')
m' = (y1 + y2)/2 = p
m = m' + p/2 = 3p/2
过M的AB的中垂线斜率为k' = -1/k = -1
中垂线：y - p = -(x - 3p/2)
过(0, 2): 2 - p = 3p/2, p = 4/5