(1) P(x, y), C(4, 0), Q(1, y)
向量PC = (4-x, 0-y) = (4-x, -y)
向量PQ = (1-x, 0)
2向量PQ = (2-2x, 0)
向量PC+2向量PQ = (4 - x +2 -2x, -y + 0) = (6 - 3x, -y)
向量PC- 2向量PQ = (4 - x - 2 + 2x, -y - 0) = (2 + x, -y)
(6-3x)(2+x) + (-y)^2 = 0
3x^2 -y^2 = 12
(2) 3x^2 -y^2 = 12, y = kx +1
解得A( [k - √(39-12k^2)]/(3-k^2), [3 - √(39-12k^2)]/(3-k^2) ) (k^2 ≠ 3)
B([k + √(39-12k^2)]/(3-k^2), [3 + √(39-12k^2)]/(3-k^2) )
AB的中点为圆心E(k/(3-k^2), 3/(3-k^2))
AB为直径d, d^2 = { [k - √(39-12k^2)]/(3-k^2) - k + √(39-12k^2)]/(3-k^2) }^2 +
{ [3 - √(39-12k^2)]/(3-k^2) - [3 + √(39-12k^2)]/(3-k^2) }^2
= 4(1+k^2)(39-12k^2)/(3-k^3) (1)
= 4r^2
r^2 = (1+k^2)(39-12k^2)/(3-k^3)
ED为半径r, r^2 = [k/(3-k^2) - 0]^2 + [ 3/(3-k^2) +2]^2 = (4k^4-35k^2+81)/(3-k^2)^2 (2)
由(1)(2),
(1+k^2)(39-12k^2) = 4k^4-35k^2+81
8k^4 -31k^2 + 21 = 0
(8k^2 - 7)(k^2 -3) = 0
k^2 = 7/8 或 k^2 = 3 (舍去)
k = ±√(7/8) = ±(√14)/4
