Question

Let $z_1$ 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 $z_2$ form 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 $a^2=3b$

As $z_1,z_2$ are roots of $z^2+az+b$

$\therefore z_1+z_2=-a,z_1{z}_2=b$

Again $0,z_1,z_2$ are the vertices of an equilateral triangle.

$\therefore 0+{z_1}^2+{z_2}^2=0z_1+z_1{z}_2+z_2.0$

$\Rightarrow {z_1}^2+{z_2}^2=z_1{z}_2$

$\Rightarrow {z_1}^2+{z_2}^2+2z_1{z}_2=z_1{z}_2+2z_1{z}_2$

$\Rightarrow {(z_1+z_2)}^2=3z_1{z}_2$

$\Rightarrow a^2=3b$