Question

Let $z=x+iy$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation $\overline{z}{z}^3+z{\overline{z}}^3=350$ is

• A. $48$
• B. $32$
• C. $40$
• D. $80$

Answer 1

The correct option is A $48$

$\overline{z}{z}^3+z{\overline{z}}^3=350$
$z\overline{z}({\overline{z}}^2+z^2)=350$

Put $z=x+iy$

$(x^2+y^2)(x^2-y^2)=175$

Since x and y are integer so only possibility is

$x^2+y^2=25$

and
$x^2-y^2=7$

so solving these equation we get

$x=\pm 4,y=\pm 3$

Thus required area $=8\times 6=48sq.unit.$