Question

If `(x+iy)^3` is an imaginary number and `|x+iy|=8`, find `x` and `y`?

Answer 1

`x=+-4sqrt3 and y=+-4`

Explanation:

`(x+iy)^3`
`=x^3+3x^2iy+3x(iy)^2+(iy)^3`
`=x^3+3x^2iy-3xy^2-iy^3`
`(x^3-3xy^2)+i(3x^2y-y^3)`
As the value is purely imaginary hence
`(x^3-3xy^2)=0`
`x^2=3y^2`
It's also given
`x+iy=8`
`=>।x+iy।=8`
`sqrt(x^2+y^2)=8`

i.e. `x^2+y^2=64` or `3y^2+y^2=64`

or `4y^2=64` i.e. `y=+-4`

and `x^2=48` i.e. `x=+-4sqrt3`
From those equations we get the answer.