(1)
C(0,4),OC = 4
tan∠CBO = OC/OB = 4/OB = 2,OB = 2,B(2,0)
代入抛物线:0 = 4a + 4a + 4,a = -1/2
y = -x²/2 - x + 4
(2)
A(-4,0)
BC = √(2² + 4²) = 2√5
P所经过的路径为三角形ABC的中位线,长度为底BC的一半,即√5
tan∠BAC = OC/AO = 4/4 = 1
∠BAC = 45˚
∠EPF = ∠DPF + ∠DPE
= 180˚ - 2∠PDF + 180˚ - 2∠PDE (PD = PF,PE = PD,∠PED = ∠PDE,∠PFD = ∠PEF)
= 2(180˚ - ∠PDE - ∠PDF)
= 2(∠CDE + ∠CDF)
= 2(90˚ - ∠ABD + 90˚ - ∠ACD)
= 2(180˚ - ∠ABD - ∠ACD)
= 2∠BAC
= 2*45˚
= 90˚
BC的方程:x/2 + y/4 = 1
令AD斜率为k (0 < k < 1),AD的方程:y = k(x + 4)
联立得交点D(4(1 - k)/(k + 2),12k/(k + 2))
F(4(1 - k)/(k + 2),0)
AC斜率为1,DE斜率为-1,DE的方程:y - 12k/(k + 2) = -[x - 4(1 - k)/(k + 2)]
AC的方程:x/(-4) + y/4 = 1
联立得交点E(2(k - 1)/(k + 2),6(k + 1)/(k + 2))
EF² = [2(k - 1)/(k + 2) - 4(1 - k)/(k + 2)]² + [6(k + 1)/(k + 2) - 0]²
= 72(k² + 1)/(k + 2)²
令f(k) = EF²/72 = (k² + 1)/(k + 2)²
f'(k) = [2k(k + 2)² - (k² + 1)*2(k + 2)]/(k + 2)⁴
= (2k² + 3k - 2)/(k + 2)⁴
= (2k - 1)/(k + 2)³ = 0
k = 1/2
EF²的最小值:72(1/2² + 1)/(1/2 + 2)² = 72/5
EF的最小值:√(72/5) = (6√10)/5