Question

The area of the figure formed by the points corresponding to the complex numbers which are the roots of the equation $1+z+z^3+z^4=0$ , is

• A. $\frac{9}{4}$ sq. units
• B. $\frac{3\sqrt{3}}{4}$ sq. units
• C. $\frac{3}{4}$ sq. units
• D. None of these

Answer 1

The correct option is B $\frac{3\sqrt{3}}{4}$ sq. units

We can factorise the given expression as :
$(z^3+1)(z+1)=0$
Hence,
$(z+1{)}^2(z^2-z+1)=0$.
$\Rightarrow z=-1$ , $z=-\omega$ , $z=-{\omega }^2$.
The three points on the Argand plane represent an equilateral triangle. The length of the side of the equilateral triangle is $\sqrt{3}$.
Area of the triangle $=\frac{1}{2}\times \sqrt{3}\times \sqrt{3}\times \frac{\sqrt{3}}{2}$
Area $=\frac{3\sqrt{3}}{4}$.
Hence, option B is correct.