Question

The roots of the equation $t^3+3a{t}^2+3bt+c=0$ are $z_1,z_2,z_3$ which represent the vertices of an equilateral triangle, then

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

Answer 1

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

$S_1=\sum z_1=-3a,S_2=\sum z_1{z}_2=3b$
since the triangle is equilateral , we have
$\sum z_1^2=\sum z_1{z}_2$
$\Rightarrow (\sum z_1{)}^2-2\sum z_1{z}_2=\sum z_1{z}_2\Rightarrow (\sum z_1{)}^2=3\sum z_1{z}_2$
$\Rightarrow (-3a{)}^2=3(3b)\Rightarrow 9a^2=9b$
$\Rightarrow a^2=b$