Question

过点A(1,2)任做两条互相垂直的的直线,分别交x,y轴于点M,N两点,求线段MN的中点P的轨迹方程.

Answer 1

设第一条直线的斜率为k,则第二条直线的斜率为-1/k

二条直线方程为y - 2 = k(x - 1)和 y - 2 = -(x - 1)/k

(1)分别取y = 0和x = 0 可得M(1 - 2/k,0),N(0,2 + 1/k)

P(1/2 - 1/k,1 + 1/(2k))

设P(x,y),x = 1/2 - 1/k (1)

y = 1 + 1/(2k),2y = 2 + 1/k (2)

(1)+(2):x + 2y = 5/2

点P的轨迹方程为:2x + 4y = 5

(2) 分别取x = 0和y = 0 可得N(0,2-k),M(1 +2k,0)

P(1/2 +k,1 - k/2)

设P(x,y),x = 1/2 +k (1)

y = 1 - k/2,2y = 2 - k (2)

(1)+(2):x + 2y = 5/2

2x + 4y = 5