Question

高二数学题已知椭圆C：4x^2+y^2=1及直线l：y=x+m,m∈R求直线l被椭圆C截得的弦的中点的轨迹.

Answer 1

设中点坐标(x,y)

y=x+m代入4x^2+y^2=1

4x^2+(x+m)^2=1

5x^2+2mx+m^1-1=0

x1+x2=-2m/5

x1,x2在直线l:y=x+m上

y1+y2=x1+x2+m+m=2m-2/5m=8/5m

4(x1+x2)+y1+y2=-8/5m+8/5m=0

x=(x1+x2)/2

y=(y1+y2)/2

8x+2y=0

y=-4x