己知动圆C过定点F(0,1),且与直线l1y=-1相切,圆心C的轨迹为E
问题描述：
己知动圆C过定点F(0,1),且与直线l1y=-1相切,圆心C的轨迹为E
己知直线l2交轨迹与PQ,且PQ中点的纵坐标为2,则|PQ|的最大值为多少
答
设C(a, b), 圆半径r= b -(-1) = b + 1
圆方程:(x - a)² + (y - b)² = (b + 1)²
过定点F(0, 1): a² + (1 - b)² = (b +1)²
a² = 4b
将a, b分别换为x, y, 圆心C的轨迹为E: x² = 4y, y = x²/4
设P(p, p²/4), Q(q, q²/4)
PQ中点的纵坐标为2: (p²/4 + q²/4)/2 = 2, p² + q² = 16(1)
|PQ|² = (p - q)² + (p²/4 - q²/4)²
= (p - q)²[1 + (p + q)²/16]
= (p² + q² - 2pq)[1 + (p² + q² + 2pq)/16]
= (16 - 2pq)(2 + pq/8)
= (1/4)(8 - pq)(16 + pq)
= (1/4)[144 - (pq + 4)²]
pq= -4时, |PQ|²最大, = 144/4 = 36
|PQ|的最大值为6