If z1, z2 and z3 are complex numbers such that |z1|=|z2|=|z3|=1 and z1+z2+z3=0, then |(1)/(z1)+(

step 1 Here in this problem we have to choose the correct option and we are given if Z1, Z2 and Z3 are

If z1, z2 and z3 are complex numbers such that...

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1$ and $z_{1}+z_{2}+z_{3}=0$, then $\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}\right|$ is (a) equal to 0 (b) equal to 1 (c) greater than 1 and less than 3 (d) equal to 3

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1$ and $z_{1}+z_{2}+z_{3}=0$, then $\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}\right|$ is
(a) equal to 0
(b) equal to 1
(c) greater than 1 and less than 3
(d) equal to 3

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Key Concepts

Symmetry in the Sum of Complex Numbers

When several complex numbers on the unit circle sum to zero, it indicates a symmetric distribution around the circle. Such symmetry is crucial in deducing further properties of the numbers, particularly when evaluating expressions involving their reciprocals or conjugates, as the same symmetry leads to cancellation effects.

Complex Numbers on the Unit Circle

Complex numbers with a modulus of 1 lie on the unit circle in the complex plane. This fact implies that these numbers have a polar representation where the magnitude is fixed at 1 and only the argument varies, which imposes special geometric properties that are often exploited in problems involving symmetry and rotation.

Reciprocal and Conjugate Relationship for Unit Complex Numbers

For any complex number z with |z| = 1, its reciprocal is equal to its complex conjugate. This relationship, 1/z = z?, is a direct consequence of the property that z multiplied by its conjugate gives |z|^2, which equals 1 when z is on the unit circle. This fact simplifies many algebraic manipulations.

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