If x, y R and x^2+y+4 i and -3+x^2 y i are conjugates to each other, then (|x|+|y|)^2=

Step-by-step solution

x^2+y+4 i and -3+x^2 y i are conjugate. Therefore, x^2+y+4 i=-3-x^2 y i (x^2+y )+4 i=(-3)-x^2 y i On comparing both sides, we get array ll & x^2+y=-3 and & 4=-x^2 y & y= 4 -x^2 =- 4 x^2 array On puting the value of y in Eq. (i), we get array ll & x^2- 4 x^2 =-3 x^4-4=-3 x^2 & x^4+3 x^2-4=0 & (x^2+4 ) (x^2-1 )=0 & x^2+4=0 & x^2=-4 & x^2-1=0 & x^2=1 x= 1 array Put the value of x= 1 in Eq. (ii), we get aligned & array l y= -4 (1)^2 =-4 x Hence, (|x|+ .|y| |^2 . =|x|^2+|y|^2+2|x||y| =(1)^2+|(-4)|^2+2|1||-4| =1+16+8=25