Question

State whether the following statement is true or false

The root of the equation $t^3+3a{t}^2+3bt+c=0$ are $z_1,z_2,z_3$ which represented the vertices of the triangle ABC on the Gaussian plane determine the centroid of the triangle. In case the triangle be equilateral $a^2=b$

• A. True
• B. False

Answer 1

The correct option is A True

Step-by-step explanation:

If $z₁,z₂$ and $z₃$ are vertices of an equilateral triangle, then we know that

$z₁²+z₂²+z₃²=z₁z₂+z₂z₃+z₃z₁.--------\left(1\right)$

Now, given $z₁,z₂$ and $z₃$ are the roots of the equation

$x³+3ax²+3bx+c=0$ ,

Hence sum of its root , $z₁+z₂+z₃=-3a,----\left(1\right)$

sum of its root taken $2$ at a time , $z₁z₂+z₂z₃+z₃z₁=3b.----\left(2\right)$

Now, consider $(z₁+z₂+z₃)²=z₁²+z₂²+z₃²+2z₁z₂+2z₂z₃+2z₃z₁.$

$=>9a²=3\left(3b\right)$

$=>a²=b.$