Assertion (A) : Let $Z_{1}, Z_{2}, Z_{3}$ be three complex number such that
$| 3 Z_{1} + 1 | = | 3 Z_{2} + 1 | = | 3 Z_{3} + 1 | a n d 1 + Z_{1} + Z_{2} + Z_{3} = 0$ then $Z_{1}, Z_{2}, Z_{3}$ will represent vertices of on equilateral triangle on the complex plane
Reason (R): $Z_{1}, Z_{2}$ 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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$B o t h A a n d R a r e t r u e a n d R i s c o r r e c t e x p l a n a t i o n o f A$

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

A is false but R is true

A is true but R is false

answer is C.

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Detailed Solution

Z₁, Z₂, Z₃ are equidistant from $(\frac{- 1}{3}, 0)$ and circumcentre of triangle is $(\frac{- 1}{3}, 0)$
also 1+ Z₁+Z₂+Z₃ = 0 $\to \frac{Z_{1} + Z_{2} + Z_{3}}{3}$ = $\frac{- 1}{3}$
centroid of the triangle is $(\frac{- 1}{3}, 0)$
Triangle is an equilateral
Z1, Z2, Z3 represents vertices of an equilateral triangle
if $Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2} = Z_{1} Z_{2} + Z_{2} Z_{3} + Z_{3} Z_{1}$
A is true R is false