参数方程与普通方程的互化问题

(1) y=cos2φ=1-2(sinφ)^2=1-2x^2
(2) x=(2-3t)/(1+t)=(5-3(1+t))/(1+t)=(5/(1+t))-3
y=(1+4t)/(1+t)=(4(1+t)-3)/(1+t)=4-(3/(1+t))
3x+5y=11
(3) x=(k+1)/(k+2)=1-(1/(k+2))
y=(2k+1)/(k+2)=2-(3/(k+2))
3x-y=1
(4) x=3t/(1+t^2)=(3t^2/(1+t^2))/t=y/t
t=y/x,代入x=3t/(1+t^2),得:
x=3(y/x)/(1+(y/x)^2)
x^2+y^2-3y=0
(5) y=sinθ^3+cosθ^3 =(sinθ+cosθ)(sinθ^2-sinθcosθ+cosθ^2)
=(sinθ+cosθ)(1-sinθcosθ)=x(1-sinθcosθ)
sinθcosθ=1-(y/x)
x^2=(sinθ+cosθ)^2=sinθ^2+cosθ^2+2sinθcosθ=1+2sinθcosθ=1+2(1-(y/x))
x^3-3x+2y=0
(6) x=a(1-t^2)/(1+t^2)=(2a/(1+t^2))-a=(y/t)-a
t=y/(x+a),代入y=2at/(1+t^2),得:
y=(2ay/(x+a))/(1+(y/(x+a))^2)
x^2+y^2=a^2