Question

已知直线y=x+m与椭圆x^2/4+y^2/8=1相交于两点A,B,且|AB|等于椭圆的焦距,求实数m

Answer 1

椭圆的焦距=4.

联立直线与椭圆方程得：3x^2+2mx+m^2-8=0,xA+xB=-2m/3,xAxB=(m^2-8)/3.

|AB|=√2√[(xA+xB)^2-4xAxB]=√2√[4m^2/9-(4m^2-32)/3]=4,m^2=3,m=-√3或m=√3.