Question

直线l在双曲线x^2\3-Y^2\2=1上截得弦长为4,其斜率为2,求直线l在y轴上的截距m

Answer 1

设直线为y=2x+m

代入

2x²-3(2x+m)²=6

2x²-12x²-12mx-3m²-6=0

10x²+12mx+3m²+6=0

x1+x2=-6m/5

x1×x2=(3m²+6)/10

代入弦长公式

√(1+k²)[(x1+x2) ²-4x1x2]

16=(1+2²)[36m²/25-2(3m²+6)/5]

80=36m²-30m²-60

6m²=140

m²=70/3

m=±√210/3