Question

A rectangle has vertices as the complex numbers whose real and imaginary parts are integers and satisfy the equation $z(\overline{z}{)}^3+(\overline{z})z^3=350.$ Then the area of the rectangle is

Answer 1

Let $z=x+iy$
$\text{Then}z(\overline{z}{)}^3+(\overline{z})z^3=350z(\overline{z}\left)\right(\overline{z}{}^2+z^2)=3502(x^2+y^2\left)\right(x^2-y^2)=350(x^2+y^2\left)\right(x^2-y^2)=175=25\times 7=35\times 5$

$\Rightarrow x^2+y^2=25\text{and}{x}^2-y^2=7\cdots \left(1\right)$
$\text{or}{x}^2+y^2=35\text{and}{x}^2-y^2=5\cdots \left(2\right)$
Solving $\left(1\right),$ we get
$x=\pm 4\text{and}y=\pm 3$
Equation $\left(2\right)$ does not have integer solution.
So the area formed by these complex numbers is $8\times 6=48$