Question

# Assertion :Let $$\displaystyle z_1, z_2, z_3$$ be distinct complex numbers & $$\displaystyle \omega^3 = 1, \omega \neq 1$$If $$\displaystyle z + \omega z_2 + \omega^2 z_3 = 0$$ then $$\displaystyle z_1, z_2, z_3$$ are the vertices of an equilateral triangle. Reason: If $$\displaystyle z_3 - z_1 = (z_2 - z_1)e^{-1^{\pi}/3}$$ then $$\displaystyle z_1, z_2, z_3$$ are vertices of an equilateral triangle.

A
Both (A) & (R) are individually true & (R) is correct explanation of (A).
B
Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
C
(A) is true (R) is false.
D
(A) is false (R) is true.

Solution

## The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).Subtracting $$\displaystyle z_1 (1 + \omega + \omega^2) = 0$$from $$\displaystyle z_1 + z_2 \omega + \omega^2z_3 = 0$$ we get$$\displaystyle (z_2 - z_1) + \omega(z_3 - z_1) = 0$$or $$\displaystyle z_3 - z_1 = -\omega^2(z_2 - z_1)$$$$\displaystyle = e^{i(\pi+\frac {4\pi}{3})-(z_2-z_3)}$$or $$\displaystyle z_3 - z_1 = (z_2 - z_1)e^{i\frac {\pi}{3}}$$$$\displaystyle \therefore z_1, z_2, z_3$$ from an equilateral triangle.Maths

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