Assertion: (A) Let $Z_{1}, Z_{2}, Z_{3}$ be the three complex numbers such that $| 3 Z_{1} + 1 | = | 3 Z_{2} + 1 | = | 3 Z_{3} + 1 |$ and $1 + Z_{1} + Z_{2} + Z_{3} = 0, t h e n Z_{1}, Z_{2}, Z_{3}$ will represent vertices of an equilateral triangle on the complex plane.
Reason: (R) $Z_{1}, Z_{2} a n d Z_{3}$ 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 A and R are true and R is the correct explanation of A

Both A and R are true and R is not correct explanation of A

A is true but R is false

A is false but R is true

answer is C.

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Detailed Solution

$Z_{1}, Z_{2}, Z_{3}$ are equidistance from $(- \frac{1}{3}, 0)$ and circumcenter of the triangle $(- \frac{1}{3}, 0)$ .
Also $1 + z_{1} + z_{2} + z_{3} = 0 \Rightarrow \frac{z_{1} + z_{2} + z_{3}}{3} = - \frac{1}{3}$
centroid is $(- \frac{1}{3}, 0)$
Triangle is equilateral
$Z_{1}, Z_{2}, Z_{3}$ represents the vertices of the equilateral triangle $\Rightarrow {z_{1}}^{2} + {z_{2}}^{2} + {z_{3}}^{2} = z_{1} . z_{2} + z_{2} . z_{3} + z_{3} . z_{1}$