Complex Numbers
Equilateral triangle in complex plane
nta_pyq_2025_apr
Grade 12
Question:
If z, z, z$\i_n C are the vertices of a_n equilateral triangle, whose centroid is z, then$$\sum 1 2 3 0 3$k=1$($zk -$z_{0}$$) 2 is equal to$
Step-by-Step Solution
Key Concept: Use the centroid relation$z_1+z_2+z_3=3z_0$and$equilateral-triangle$rotation by cube roots of unity.
$z + z + z = 3z$1 2 3 0 2 2$(1)$($z_{1} +$$z_{2}$$+ z_{3}$) = 9z 0 2 2 2 2 2 2 2 $\Rightarrow$$z + z + z + 2$($z + z + z$) = 9z 1 2 3 1 2 3 0 2 2 2 2 $\Rightarrow$$z + z + z = 3z$1 2 3 6 3 2 2 2 2$\sum ($zk -$z_{0}$$) = ($$z_{1}$-$z_{0}$$) + ($$z_{2}$-$z_{0}$$) + ($$z_{3}$-$z_{0}$$)$k=1$2 2 2$2 = z + z + z + 3z - 2$($$z_{1}$+$z_{2}$+$z_{3}$$)$$z_{0}$$1 2 3 0 2$2 = 6z - 6z$0 0 = 0$
Correct Answer: 1