Question

If $\left(2x+\frac{y}{8},2x-\frac{y}{8}\right)=xy$, then show that $f(m,n)+f(n,m)=0$

Answer 1

Let , $X=2x+\frac{y}{8}$ ...........1

$Y=2x-\frac{y}{8}$ ............2

from eq. 1 and 2

$x=\frac{X+Y}{4}$ and $y=4(X-Y)$

so,

$f(X,Y)=\frac{X+Y}{4}4(X-Y)$

$f(X,Y)=X^2-Y^2$

Similarly ,

$f(m,n)=m^2-n^2$ ...............3

$f(n,m)=n^2-m^2$ .................4

Add 3 and 4

$f(m,n)+f(n,m)=m^2-n^2+n^2-m^2=0$