Question

已知三角形ABC,AB两点在椭圆x2+3y2=4上,C点在直线L上,y=x+2,AB平行于L线,求AB中点的轨迹

Answer 1

AB:y=x+m

椭圆方程x2+3y2=4与直线y=x+m联立

得到4x^2+6mx+3m^2-4=0

x1+x2=-3m/2,

A(x1,y1),B(x2,y2)

AB的中点P(x,y),

x=(x1+x2)/2,y=(y1+y2)/2

x=-3m/4,

y1=x1+m,y2=x2+m

y=(y1+y2)/2

=(x1+x2)/2+m

=m/4

x/y=-3,

y=-x/3