Question

if $z_1$ and $z_2$ be two roots of the equation $z^2+az+b=0,z$ being complex. Further,assume that the origin $z_1$ and $z_2$ from an equilateral triangle, then

• A. $a^2=b$
• B. $a^2=2b$
• C. $a^2=3b$
• D. $a^2=4b$

Answer 1

The correct option is C $a^2=3b$

Given that$:z_1$ and $z_2$ are two roots of the equation $z_2+az+b=0,z$ being complex.

$\Rightarrow$ We have $z_1+z_2=-a$

$z_1{z}_2=b$ (Sum and Product of roots)

$\Rightarrow z_2=z_1\left(\cos {60}^{\circ }\right)+i\left(\sin {60}^{\circ }\right)=z_1(\frac{1}{2}+\frac{\sqrt{3}}{2}i)$

$\therefore 2z_2-z_1=\sqrt{3}i{z}_1$

$⟹(2z_2-z_1{)}^2=-3z_1^2$

Hence, $(z_1^2+z_2^2)=z_1{z}_2$

$⟹a^2-2b=b$ or $a^2=3b$