If $x^4 + y^4 + x^2y^2 = 117 $ and $x^2 + y^2 - xy = 3( 4 + \sqrt{3})$, then the value of $(x^2+y^2)$ will be :

12

If $x^4 + y^4 + x^2y^2 = 117 $

$x^2 + y^2 - xy = 3( 4 + \sqrt{3})$,------ (A)

then the value of $(x^2+y^2)$ = ?

We know that,

x⁴ + x²y² + y⁴ = (x² – xy + y²) (x² + xy + y²)

In this type of questions,

if $x^2 + y^2 - xy = 3( 4 + \sqrt{3})$,

then, $x^2 + y^2 - xy = 3( 4 - \sqrt{3})$, ( Always),------ (B)

So add both the equations,

2 ( x² + y² ) = 24

( x² + y² ) = 12