Question

已知抛物线c:y^2=ax的焦点与双曲线x^2/2-y^2/2=1的右焦点重合

(1)求抛物线c的方程：

(2)过点A(2.0)作倾斜角为π/4的直角，与抛物线c交于M,N两点，判断角MON是否为直角。若角MON为直角，请给出证明：若不是直角，请说明理由

Answer 1

(1) 显然 a > 0, 抛物线焦点(a/4, 0)
双曲线c = √(2 + 2) = 2, 右焦点(2, 0)
a/4 = 2, a = 8
抛物线: y² = 8x

(2)过点A(2.0)作倾斜角为π/4的直线:
y - 0 = (x - 2)tan(π/4)
y = x - 2
(x - 2)² - 8x = 0
x² - 12x + 4 = 0
x₁,₂ = 6 ± 4√2, y₁,₂ = 4 ± 4√2
kOM*kON = y₁y₂/(x₁x₂) = -4 ≠ -1
不是直角