Question

Let $z=x+iy$ be a complex number where $x$ and $y$ are integers, then the perimeter of the rectangle whose vertices are represented by the roots of $z{\overline{z}}^3+\overline{z}{z}^3$ = 1088 is

• A. $16$ units
• B. $32$ units
• C. $28$ units
• D. None of these

Answer 1

The correct option is B $32$ units

Given $z{\overline{z}}^3+\overline{z}{z}^3=1088z\overline{z}\{{\overline{z}}^2+z^2\}=1088{|z|}^2\{{\overline{z}}^2+z^2\}=1088$

$(x^2+y^2)\left\{2\right(x^2-y^2\left)\right\}=1088(x^2+y^2)(x^2-y^2)=554$

Now since $x,y$ are integers, so we can conclude that $x^2=5^2$ and $y^2=3^2$

$\therefore$ vertices of the rectangle are $(5,3),(5,-3),(-5,3)$ and $(-5,-3)$ and perimeter $=32$ units