Statement I : Let $z_{1}, z_{2}$ and $z_{3}$ be three complex numbers, such that $| 3 z_{1} + 1 | = | 3 z_{2} + 1 | = | 3 z_{3} + 1 |$ and $1 + z_{1} + z_{2} + z_{3} = 0$ , then $z_{1}, z_{2}, z_{3}$ will represent vertices of an equilateral triangle on the complex plane.
Statement II: $z_{1}, z_{2}$ and $z_{3}$ represent vertices of an equilateral triangle, if $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1} = 0$ .

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Both statement I and statement II are false

Both statement I and statement II are true

Statement I is incorrect but statement II is true

Statement I is correct but statement II is false

answer is C.

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Detailed Solution

We have, $| 3 z_{1} + 1 | = | 3 z_{2} + 1 | = | 3 z_{3} + 1 |$
$∴ z_{1}, z_{2}$ and $z_{3}$ are equidistant from $(- \frac{1}{3}, 0)$ and circumcentre of triangle is $(- \frac{1}{3}, 0)$ .
Also, $1 + z_{1} + z_{2} + z_{3} = 0$
$\Rightarrow \frac{1 + z_{1} + z_{2} + z_{3}}{3} = 0$
$\Rightarrow \frac{z_{1} + z_{2} + z_{3}}{3} = - \frac{1}{3}$
$∴$ Centroid of the triangle is $(- \frac{1}{3}, 0)$ .
So, the circumcentre and centroid of the triangle coincide.
Hence, required triangle is an equilateral triangle.
Therefore, statement I is true. Also, $z_{1}, z_{2}$ and $z_{3}$ represent vertices of an equilateral
triangle, if $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} - (z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}) = 0$ .
Therefore, statement II is false.