Question

if(x+iy)³=y+vi then show that(y/x+v/y)=4(x²-y²)​

Answer 1

Answer:

Given, (x+iy) 3

=u+ivx 3+(iy) 3 +3.x.iy(x+iy)=u+iv

x 3 +i 3 y 3+3x 2 yi+3xy 2 i 2

=u+ivx 3−iy 3 +3x 2 yi−3xy 2

=u+iv(x 3−3xy 2)=i(3x 2 y−y 3 )=u+iv

On equating real and imaginary parts, we get

u=x 3−3xy 2,v=3x 2y−y 3

∴ xu+ yv = xx 3 −3xy

2 + y3x 2 y−y 3

= xx(x 2 −3y 2 ) + yy(3x 2−y 2 )

=x 2 −3y 2+3x 2−y 2=4x 2−4y 2=4(x 2 −y 2 )