Question

已知实数x,y满足(x-2)^2+(y-1)^2=1,求z=((y+1)/x的最大值与最小值

Answer 1

令x=2+sina,y=1+cosa

令(y+1)/x=(cosa+1)/(sina+2)=A

Asina+2A=cosa+1

Asina-cosa=1-2A

√(A^2+1)sin(a+b)=1-2A

sin(a+b)=(1-2A)/√(A^2+1)

由-1≤sin(a+b)≤1,得

-1≤(1-2A)/√(A^2+1)≤1

(1-2A)^2/(A^2+1)≤1

A(3A-4)≤0

0≤A≤4/3

z=(y+1)/x的最大值为4/3,最小值为0.