参化普：x=t+1/t,y=t-1/t（t为参）第二个：x=e^t+e^-ty=2(e^t-e^-t)（t为参）

Question

参化普：x=t+1/t,y=t-1/t（t为参）第二个：x=e^t+e^-t y=2(e^t-e^-t)（t为参）

Answer 1

(1) 二者相加：x + y = 2t,t = (x+y)/2
代入x的参数方程：x = (x+y)/2 + 2/(x+y) = [(x+y)² + 4]/[2(x+y)]
2x² + 2xy = x² + 2xy + y² + 4
x² - y² = 4
(2) (1)+(2)/2:x + y/2 = 2e^t
e^t = x/2 + y/4
代入x的参数方程:x = x/2 + y/4 + 1/(x/2 + y/4)
x/2 - y/4 = 1/(x/2 + y/4)
(x/2)² - (y/4)² = 1
显然x = e^t + e^(-t) ≥ 2