作业帮大题 圆锥曲线
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大题 圆锥曲线
大题 圆锥曲线
大题 圆锥曲线
(1)
设OP与+x间的角为θ,P((√7cosθ)/2,(√7sinθ)/2)
F1(-c,0),F2(c,0)
向量PF1 = (-c - (√7cosθ)/2,-(√7sinθ)/2)
向量PF2 = (c - (√7cosθ)/2,-(√7sinθ)/2)
向量PF1•向量PF2 = (7/4)cos²θ - c² + (7/4)sin²θ
= (7/4)(sin²θ + cos²θ) -c²
= 7/4 - c² = 3/4
c² = 1,c = 1
e = c/a = 1/a = √2/2,a = √2
b = a² - c² = 2 - 1 = 1
x²/2 + y² = 1
(2)
y = kx - 1/3
x²/2 + (kx - 1/3)² = 1
9(2k² + 1)x² - 12kx -16 = 0
x₁,₂ = [2k ± 2√(9k² + 4)]/[3(2k² + 1)]
yx₁,₂ = kx₁,₂ - 1/3
AB的中点即圆心C(2k/[3(2k² + 1)],-1/[3(2k² + 1)])
直径D,D² = (x₁ - x₂)² + (y₁ -y₂)² = (x₁ - x₂)² + (kx₁ - 1/3 - x₂ + 1/3)²
= (k² + 1)(x₁ - x₂)²
= 16(k² + 1)(9k²+ 4)/[9(2k² + 1)²]
r² = 4(k² + 1)(9k²+ 4)/[9(2k² + 1)²]
圆的方程:[x - 2k/(6k² + 3)]² + [y + 1/(6k² + 3)]² = 4(k² + 1)(9k²+ 4)/[9(2k² + 1)²]
总过(0,1)