Question

Given that the complex numbers which satisfy the equation $|z{\overline{z}}^3|+|\overline{z}{z}^3|=350$ form a rectangle in the Argand plane with the length of its diagonal having an integral number of units, then

• A. Area of rectangle is 48 sq. units
• B. if $z_1,z_2,z_3,z_4$ are vertices of rectangle, then $z_1+z_2+z_3+z_4=0$
• C. Rectangle is symmetrical about the real axis
• D. $arg(z_1-z_3)=\frac{\pi }{4}$ or $\frac{3\pi }{4}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A Area of rectangle is 48 sq. units
B if $z_1,z_2,z_3,z_4$ are vertices of rectangle, then $z_1+z_2+z_3+z_4=0$
C Rectangle is symmetrical about the real axis

Let $z=x+iy$ where x, y satisfy the given equation. Hence, $(x^2+y^2)(x^2-y^2)=175$
$\Rightarrow x^2+y^2=25$ and $x^2-y^2=7$ (as all other possibilities will give non-integral solutions)
Hence, possible values of z will $4+3i,4-3i,-4+3i$, and $-4-3i$. Clearly, it will form a rectangle having length of the diagonal $10$.
From the diagram, options a,b,c are correct.