Question

If the four roots of the equation $z^4+z^3+2z^2+z+1=0$ forms a quadrilateral on the argand plane, then the area of quadrilateral is-

• A. $\frac{\sqrt{3}+2}{8}$
• B. $\frac{\sqrt{3}+2}{4}$
• C. $\frac{\sqrt{3}+2}{2}$
• D. $\frac{\sqrt{3}+2}{3}$

Answer 1

Answer: (B)

$\frac{\sqrt{3}+2}{4}$

Given, $z^4+z^3+2z^2+z+1=0$

Dividing the equation by $z^2$

$z^2+\frac{1}{z^2}+z+\frac{1}{z}+2=0$

${(z+\frac{1}{z})}^2+(z+\frac{1}{z})=0$

$\Rightarrow$ $z+\frac{1}{z}=0z+\frac{1}{z}+1=0$

$\Rightarrow$ $z=\pm i$ or $z=w,w^2$

Plotting solution on Argand plane

Required area $=\frac{1}{2}(2+\sqrt{3})\times \frac{1}{2}=\frac{\sqrt{3}+2}{4}$