If ( x+ y= 3 +1 2 ) and ( x+ y= ) ( 3 -1 2 ), then ( ^2 ( x-y 2 )+ ^2 ( x+y 2 )= )

Step-by-step solution

It is given that ( array ll x+ y & = 3 +1 2 2 & 2 x+y 2 x-y 2 = 3 ^2+1 2 array ) ( x+y 2 x-y 2 = 3 +1 4 ) ...(i) and ( x+ y= 3 -1 2 ) ( 2 x+y 2 x-y 2 = 3 -1 2 ) ( x+y 2 x-y 2 = 3 -1 4 ) ...(ii) On dividing relation (i) and (ii), we get ( aligned & x+y 2 = 3 +1 3 -1 & ^2 x+y 2 = 3+1+2 3 3+1-2 3 = 2+ 3 2- 3 & ^2 (x+y) 2 = 4+3+4 3 4-3 =7+4 3 (iii) aligned ) On squaring and adding Eqs. (i) and (ii), we get ( aligned & ^2 x-y 2 = 1 16 2(3+1)= 1 2 ^2 x-y 2 =2 & ^2 x-y 2 = ^2 x-y 2 -1=2-1=1 aligned ) So, ( ^2 x-y 2 + ^2 x