已知椭圆c1的离心率为根号3/2,抛物线c2：x^2=4y的焦点在椭圆的顶点上（1）过A(0,1)
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(1) e² = c²/a² = (a² - c²)/a² = (4 - b²)/4 = 3/4,b = 1

C2为开口向上的抛物线,焦点只能是椭圆的上顶点(0,1),p/2 = 1,p = 2

C2:x² = 4y

(2)F(0,1/2),p/2 = 1/2,p = 1

x² = 2y,y = x²/2

设P(u,u²/2)

y' = x

过点p的切线:y - u²/2 = u(x - u),y = ux - u²/2

代入x²/4 + y²= 1

(4u² + 1)x² - 4u³x + u⁴ - 4 = 0

x₁ + x₂ = 4u³/(4u² + 1)

x₁x₂ = (u⁴ - 4)/(4u² + 1)

y₁,₂ = ux₁,₂ - u²/2

设OA,OB的斜率分别为m,n

mn = -1 = y₁y₂/(x₁x₂)

-x₁x₂ = y₁y₂

-(u⁴ - 4)/(4u² + 1) = (ux₁ - u²/2)(ux₂ - u²/2) = u²x₁x₂ - (u³/2)(x₁ + x₂) + u⁴/4

= u²(u⁴ - 4)/(4u² + 1) - (u³/2)*4u³/(4u² + 1) + u⁴/4

5u⁴ - 16u² - 16 = 0

(5u² + 4)(u² - 4) = 0

u² = 4

u = ±2

P(±2,2)

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