Find all real pairs (x,y) such that: |x+y|+|x-y|=2 ?

if x+y and x-y >0 then this is x=1, y is in range of [-1,1]

if x+y <0 and x-y <0 then x is -1 , y is in range of [-1,1]

if x+y <0 x-y>0 y=-1 x is range of [-1,1]

if x+y >0 x-y <0 then y=1 x is in range of [-1,1]

so it is a square of size 2 centered at origin.

Given

|x+y|+|x-y| = 2 --------------- (1)

Let

|x+y|-|x-y| = k ----------------- (2)

multiplying (1) and (2)

(x+y)² - (x-y)² = 2k or

4xy = 2k or xy = k/2 ---------------- (3)

adding (1) and (2) we get

2*|x+y| = 2+k

4x² + 4y² + 8xy = 4+k² + 2k

Using (3) we get

4x² + 4y² + 4k = 4+k² + 2k or

x² + y² = {(k² -2k +4)/4} or

x² + y²+2xy= {(k² -2k +4)/4} +k (using (3)or

(x+y)² = {(k² +2k +4)/4} and (x-y)² = {(k² -6k +4)/4}

Or from (1)

{√(k² +2k +4)}+ {√(k² -6k +4)} = 4 or

{√(k² +2k +4)} = 4 - {√(k² -6k +4)}, squaring both sides we get

(k² +2k +4) = 16 + (k² -6k +4) -8 {√(k² -6k +4)} or

8 {√(k² -6k +4)} = 20-8k or

4(k² -6k +4) =(5-2k)² or

4k² -24k +16 = 25+4k² -20k or

4k = -9 or k = -9/4 or

x² + y² = {(k² -2k +4)/4} = {(81/16) +(18/4) +4}/4 = 169/64 and xy = -9/8 gives

(x+y)² = 25/64 or x+y = +/- 5/8 similarly

(x-y)² = 313/64 or x-y = +/- (√313)/8 or

x = 0.6*[+/- 5/8+/- (√313)/8] and y = x = 0.6*[+/- 5/8-/+ (√313)/8]

(1) x + y ≥ 0 and x - y ≥ 0 .....right

x = 1 .....- 1 ≤ y ≤ 1

(2) x + y ≤ 0 and x - y ≤ 0 .....left

x = - 1 .....- 1 ≤ y ≤ 1

(3) x + y ≤ 0 and x - y ≥ 0 .....lower

y = - 1 .....- 1 ≤ x ≤ 1

(4) x + y ≥ 0 and x - y ≤ 0 .....upper

y = 1 .....- 1 ≤ x ≤ 1

X+Y+X-Y=2

2X=2

X=1

Anypoint on X=1 would work e.g. (1,0) (1,5) (1,99999999)