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  4. If the complex numbers z1, z2 and z3 represent the vertices of an equilateral triangle such that |z1|=|z2|=|z3|, then

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Q. If the complex numbers $z_1, z_2$ and $z_3$ represent the vertices of an equilateral triangle such that $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|$, then

Complex Numbers and Quadratic Equations

Solution:

Let $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=k$ (say),
$\Rightarrow z_1, z_2, z_3$ lie on a circle with centre at the origin and radius $k$.
As $z_1, z_2, z_3$ are vertices of an equilateral triangle, the circumcentre and the centroid of the triangle coincide. Therefore,
$\frac{1}{3}\left(z_1+z_2+z_3\right)=0 \Rightarrow z_1+z_2+z_3=0$