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Question: If z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub> are vertices of an equilateral triangle with z<sub>0<...

If z1, z2, z3 are vertices of an equilateral triangle with z0 its circumcentre, then z12 + z22 + z32 is equal to-

A

z02

B

9z02

C

3z02

D

2z02

Answer

3z02

Explanation

Solution

Sol. If z12 + z22 + z32 = z1z2 + z2z3 + z3z1 ….(1)

(z1 + z2 + z3)2 = ∑z12\sum_{}^{}{z_{1}}^{2}+2 ∑z2z3\sum_{}^{}{z_{2}z_{3}}

= ∑z12\sum_{}^{}{z_{1}}^{2}+ 2∑z12\sum_{}^{}{z_{1}}^{2}= 3 ∑z12\sum_{}^{}{z_{1}}^{2} (using (1))

The centroid is at thecircumcentre

z0 = z1+z2+z33\frac{z_{1} + z_{2} + z_{3}}{3} ; (z1 + z2 + z3)2 = 9z02

∑z12\sum_{}^{}{z_{1}}^{2}= 3 z02