设f(x+y,y/x)=x^2+y^2,则f(x,y)=?,f(x-y,xy)=?

题目

设f(x+y,y/x)=x^2+y^2,则f(x,y)=?,f(x-y,xy)=?

答案

设x+y=a,y/x=b,则有x=a/(b+1),y=ab/(b+1),故f(x+y,y/x)=f(a,b)=x^2+y^2=[a/(b+1)]^2+[ab/(b+1)]^2=a^2(b^2+1)/(b+1)^2即f(x,y)=x^2(y^2+1)/(y+1)^2代入x-y和xy即得f(x-y,xy)