X and Y

Squaring both side and simplifying we get

$2x^{2}$ + $2y^{2}$ = $2^{2}$

$x^{2}$ + $y^{2}$ = $2$

so for maximum value of the asked expression

$x^{2}$ + $y^{2}$ -6 $x$

so -6 $x$ is $maximum$ when $x$ = $-1$

on putting values we get $2+(-6)(-1)$ = $\boxed{8}$

$The~ equation~ leads~ to~ four~ options.\\ X+Y+X-Y=\pm~2.~~~\implies~X=\pm~1.\\ X+Y-X+Y=\pm~2.~~~\implies~Y=\pm~1.\\ What~ ever~ options~ we~ take~X^2+Y^2=2.\\ To~maximize~we~should~have~maximum~=tive~value~of~-6X.\\ This~is~possible~if~X=-1,~~and~~-6X=6,\\ Max.=2+6=\Large~~\color{#D61F06}{8}.$

We can split the part we're trying to optimize: x^2 - 6x + y^2 >>>> (x^2-6x) + y^2. The negatives of the expression became pointless because of the squared monomials except in the (-6x) part. Thus, we can optimize this by making, x , into a large negative number. It is optimized at x = -1 because any lower of a number would not allow the original equation to follow through for the sum of the absolute values would always be greater than 2.

Logically, x = - 1 and y = 1 >> maximum value = 8

|x| has property of making a number +ve when +ve no change

when -ve it multiplies -1

u will have 4 eqns like that

select the max value of x & y from them

then put it in exp the ans is 8

|x+y|+|x-y|=2

sqrt(x^2+y^2)+sqrt(x^2-y^2)=2

converting to for (a+b)^2 = a^2+b^2+2ab

we get x^2+y^2+sqrt(x^2-y^2)=2

so the only possible way is x^2+y^2=1 and sqrt(x^2-y^2)=1

solving both of them x=+ or - 1 y=+ or -1

so x=-1 and y=+ or -1 gives maximum value and it is 8.