怎么证明双曲线的渐近线

Question

怎么证明双曲线的渐近线

Answer 1

x²/a²-y²/b²=1 ①
先在第一象限讨论 渐近线方程为 y=(b/a)x ②
在同一个横坐标下,
直线相对应N点纵坐标:Y=(b/a)x,
双曲线的相对应点M纵坐标为 y²/b²=x²/a²-1 y=b/a√(x²-a²)
NM
=(b/a)x-b/a√(x²-a²)
=(b/a)[x-√(x²-a²)]
=(b/a)[x-√(x²-a²)]*[x+√(x²-a²)]/[x+√(x²-a²)]
=(b/a)*a²/[x+√(x²-a²)]
=ab/[x+√(x²-a²)]
点M(x,b/a√(x²-a²))到直线①的距离MQ为,
[bx-b√(x²-a²)]/(√b²+a²)=b[x-√(x²-a²)]/(√b²+a²)
b/a>b/(√b²+a²)
MQ