Question

Assertion :Let $z_1,z_2,z_3$ be distinct complex numbers & ${\omega }^3=1,\omega \ne 1$
If $z+\omega z_2+{\omega }^2{z}_3=0$ then $z_1,z_2,z_3$ are the vertices of an equilateral triangle. Reason: If $z_3-z_1=(z_2-z_1)e^{-1^{\pi }/3}$ then $z_1,z_2,z_3$ 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.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).

Subtracting $z_1(1+\omega +{\omega }^2)=0$
from $z_1+z_2\omega +{\omega }^2{z}_3=0$ we get
$(z_2-z_1)+\omega (z_3-z_1)=0$
or $z_3-z_1=-{\omega }^2(z_2-z_1)$
$=e^{i(\pi +\frac{4\pi }{3})-(z_2-z_3)}$
or $z_3-z_1=(z_2-z_1)e^{i\frac{\pi }{3}}$
$\therefore z_1,z_2,z_3$ from an equilateral triangle.