Y=4/x的图象绕点0旋转45度以后的图像方程?

Question

Y=4/x的图象绕点0旋转45度以后的图像方程?

Answer 1

设y=4/x 上的任意点(a,b), ab=4,它到原点距离为r, 与原点连线后, 连线与X轴形成的夹角为θ. 则
a = r cosθ
b = r sinθ
点(a,b) 旋转45度后 ,点的坐标为（x,y）
x = rcos(θ + π/4) = rcosθ cos(π/4) - rsinθsin(π/4)
= (√2 /2)(a - b)
√2x = a - b
y = r sin(θ+ π/4) = rsinθcos(π/4) + rcosθsin(π/4)
= (√2 /2) (a + b)
√2y = a + b
以上两式相加和相减：
√2y - √2x = 2 b
√2y +√2 x = 2 a
以上两式相乘：
( √2y - √2x )( √2y + √2x )=4ab=4 x 4 = 8
2y^2 - 2x^2 = 8
y^2 - x^2 = 4