...^2=9,动圆p与圆M外切并与圆N内切,圆心P的轨迹为曲线C。
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发布时间：2024-10-20 18:30
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时间：2024-12-01 04:53
2. 当圆P半径最长时, P在x轴上, P(2, 0) (圆P与圆M相切于(0, 0), 与圆N相切于(4, 0)), 半径R= 2
设L斜率为k, 方程y = kx + b, kx - y + b = 0
M与L的距离为圆M半径r = 1 = |-k - 0 + b|/√(k² + 1)
k² + 1 = k² - 2kb + b² (i)
P与L的距离为圆P半径R = 2 = |2k - 0 + b|/√(k² + 1) (ii)
(2k + b)² = 4(k² + 1) = 4k² - 8kb + 4b²
b² = 4kb
显然b ≠ 0
b = 4k
代入(i): k = ±√2/4
b = ±√2
由于对称性，不妨只考虑k > 0, b > 0
y = √2x/4 + √2 (iii)
代入x²/4 + y²/3 = 1: 7x² + 8x - 8 = 0
x₁ + x₂ = -8/7
x₁x₂ = -8/7
|AB|² = (x₁ - x₂)² + (y₁ - y₂)² = (x₁ - x₂)² + (√2x₁/4 + √2 - √2x₂/4 - √2)² = (9/8)(x₁ - x₂)² = (9/8)[(x₁ + x₂)² - 4x₁x₂]
= (9/8)[(-8/7)² - 4(-8/7)]
= 326/49
|AB| = 18/7
