Let , be the roots of the equation x^2-a x-b=0 with Im ( ) Im ( ) . Let P_n= ^n- ^n . If P ₃=-5 7 i, P ₄=-3 7 i, P ₅=11 7 i and P ₆=45 7 i , then | ^4+ ^4 | is equal to .

Step-by-step solution

aligned & + = a =- b & P ₆= aP ₅+ bP ₄ & 45 7 i = a 11 7 i + b (-3 7 ) i & 45=11 a -3 ~b aligned and aligned & P₅= aP ₄+b P₃ & 11 7 i = a (-3 7 i )+ b (-5 7 i ) & 11=-3 a -5 ~b & a =3, ~b =-4 & | ^4+ ^4 |= ( ^4- ^4 )^2+4 ^4 ^4 & = -63+4.4^4 & = -63+1024 = 961 =31 aligned