Question

The origin, $z_1$ and $z_2$ are the vertices of an equilateral triangle where $z_1,z_2$ are roots of the equation $z^2+az+b=0.$ Then

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

Answer 1

Answer: (B)

$a^2=3b$

$z_1+z_2=-a$
$z_1{z}_2=b$
Now
Since $0,z_1,z_2$ are the vertices of an equilateral triangle.
$z_1^2+z_2^2+0^2=z_1\left(0\right)+z_2\left(0\right)+z_1\left(z_2\right)$
$(z_1+z_2{)}^2-2z_1{z}_2=z_1{z}_2$
$a^2=3z_1{z}_2$
$a^2=3b$