Question

过抛物线y²=4x的焦点F作互相垂直的弦AC，BD，则点A，B，C，D所构成四边形的面积的最小值为

过抛物线y²=4x的焦点F作互相垂直的弦AC，BD，则点A，B，C，D所构成四边形的面积的最小值为多少？

Answer 1

S = △ABD的面积 + △CBD的面积 = (1/2)BD*AF + (1/2)BD*CF = (1/2)BD*(AF + CF) = (1/2)BD*AC
F(1, 0)
令AC的斜率为k, BD的斜率为-1/k
AC: y = k(x - 1)
与抛物线y² = 4x联立: k²(x - 1)² - 4x = 0, k²x²燃橡 - 2(k² + 2)x + k² = 0
令其二根皮拆旁为御迅a, c, 则a + c = 2(k² + 2)/k², ac = 1
AC²= (a - c)² + [k(a - 1) - k(c - 1)]² = (k² + 1)[(a + c)² - 4ac]²
= (k² + 1){[2(k² + 2)/k²]² - 4}
= 16(k² + 1)²/(k²)²
AC = 4(k² + 1)/k²
类似地BD = 4(k² + 1)
S = (1/2) = 16(k² + 1)²/k² = 16(k + 1/k)²
k = 1/k时, S最小，此时k = ±1
其余请自己做。