Question

Given that the complex numbers which satisfy the equation $\left|z{\overline{z}}^3\right|+\left|\overline{z}{z}^3\right|=350$ form a rectangle in the Argand plane then which of the following can be true

• A. Area of rectangle is 48sq. 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 48sq. 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 $z=x+iy$.

$|z{\overline{z}}^3|+|\overline{z}{z}^3|=350$

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

$|(x^2+y^2)(x^2-2xyi-y^2+x^2+2xyi-y^2)|=350$

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

$x^2+y^2=25$---------(1)

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

Adding (1) and (2)

$2x^2=32⟹x=\pm 4$

and $y=\pm 3$

So the four vertices of the rectangle are $(4,3),(4,-3),(-4,-3)$ and $(-4,3)$

$\Rightarrow$ Hence the rectangle is symmetrical about the real axis.

$\therefore$ Area of rectangle $=length\times breadth=(4+4)\times (3+3)=48sq.units$

$\therefore$ $z_1+z_2+z_3+z_4=4+3i+4-3i-4-3i-4+3i=0$