如图,抛物线y=ax²+bx+c交x轴于A,B两点,交y轴于C点,顶点为点D,直线CD交x轴于点E,已知抛物线

(1)
tan∠CEO = OC/EO = 2/EO = 1/3
EO = 6, E(-6, 0)
对称轴为x = 1, 则B的横坐标为1 + (1 + 6) = 8, B(8, 0)
方程为y = a(x + 6)(x - 8)
其常数项为-48a = 2
a = -1/24
y = -x²/24 + x/12 + 2
CD的方程: x/(-6) + y/2 = 1, y = x/3 + 2
(2)
设Q(1, q)
OQ斜率为q
OP斜率为-1/q
OP的方程: y = -x/q
与y = x/3 + 2联立得P(-6q/(q + 3), 6/(q + 3))
OP² = 36(q² + 1)/(q + 3)²
OQ² = 1 + q²
(a) OP/OQ = 1/3
OP² = OQ²/9
36(q² + 1)/(q + 3)² = (1 + q²)/9
36/(q + 3)² = 1/9
q + 3 = ±18
q = 15或q = -21
P(-5, 1/3)或(-7, -1/3)
(b)OQ/OP = 1/3
OQ² = OP²/9
36(q² + 1)/(q + 3)² = 9(1 + q²)
(q + 3)² = 4
q + 3 = ±2
q = -1或q = -5
P(3, 3)或(-15, -3)
(3)
容易看出N(1, 1)