設 雙曲線標準方程為X^2/a^2 - Y^2/b^2 = 1(a>0,b>0)
而反比例函數(shù)的標準型是 xy = c (c ≠ 0)
但是反比例函數(shù)確實是雙曲線函數(shù)經(jīng)過旋轉(zhuǎn)得到的
因為xy = c的對稱軸是 y=x,y=-x 而X^2/a^2 - Y^2/b^2 = 1的對稱軸是x軸,y軸
所以應該旋轉(zhuǎn)45度
設旋轉(zhuǎn)的角度為 a （a≠0,順時針）
(a為雙曲線漸進線的傾斜角)
則有
X = xcosa + ysina
Y = - xsina + ycosa
取 a = π/4
則
X^2 - Y^2 = (xcos(π/4) + ysin(π/4))^2 -(xsin(π/4) - ycos(π/4))^2
= (√2/2 x + √2/2 y)^2 -(√2/2 x - √2/2 y)^2
= 4 (√2/2 x) (√2/2 y)
= 2xy.
而xy=c
所以
X^2/(2c) - Y^2/(2c) = 1 (c>0)
Y^2/(-2c) - X^2/(-2c) = 1 (c