Question

求过点M(3,1)且被点M平分的双曲线4分之x平方减y平方等于1的弦所在直线方程

Answer 1

直线经过点M(3,1),则设所求的直线方程为:

y+1=k(x-3),即y=kx-3k-1

解下方程组:

y=kx-3k-1.(1)

x^2/4-y^2=1.(2)

即可得弦与双曲线的交点坐标:

x^2/4-(kx-3k-1)^2=1

(0.25-k^2)x^2+2k(3k+1)x-(3k+1)^2-1=0

x1+x2=-2k(3k+1)/(0.25-k^2)

因为x1、x2是弦与双曲线x^2/4-y^2=1两个交点的横坐标,由已知条件可知,点M(3,-1)为弦的中点,则

(x1+x2)/2=-k(3k+1)/(0.25-k^2)=3

k=-0.75

直线方程:y=-0.75x+1.25,即3x+4y-5=0