Statement 1 z1 and z2 are two complex numbers such that |z1|=|z2|=1 . Then, |(1)/(z1)+(1)/(z2)|=

step 1 In this question, there are two statements. We need to find which statement is true and based on

Statement 1 z1 and z2 are two complex numbers such that...

Statement 1 $\mathrm{z}_{1}$ and $\mathrm{z}_{2}$ are two complex numbers such that $\left|\mathrm{z}_{1}\right|=\left|\mathrm{z}_{2}\right|=1 .$ Then, $\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=\left|z_{1}+z_{2}\right|$ and Statement 2 For any two complex numbers satisfying the conditions $\left|z_{1}\right|=\left|z_{2}\right|=r,\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=\left|z_{1}+z_{2}\right|$.

Statement 1 $\mathrm{z}_{1}$ and $\mathrm{z}_{2}$ are two complex numbers such that $\left|\mathrm{z}_{1}\right|=\left|\mathrm{z}_{2}\right|=1 .$ Then, $\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=\left|z_{1}+z_{2}\right|$
and
Statement 2 For any two complex numbers satisfying the conditions $\left|z_{1}\right|=\left|z_{2}\right|=r,\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=\left|z_{1}+z_{2}\right|$.

Key Concepts

Properties of Complex Numbers on a Circle

Complex numbers with a fixed modulus, such as those lying on the unit circle or on a circle of radius r, exhibit special symmetry and algebraic properties. These properties allow for simplifications in expressions, such as showing the equality of moduli of sums and reciprocals under specific conditions, and are pivotal in problems involving rotational invariance and polar representations.

Use of Polar Form in Complex Analysis

Expressing complex numbers in polar form, where a number is represented as re^(i?), is a powerful method that simplifies the multiplication, division, and inversion of complex numbers. This form makes it clear how the modulus and argument behave under various operations, such as finding the reciprocal or adding complex numbers with the same modulus.

Complex Number Modulus

The modulus of a complex number measures its distance from the origin in the complex plane and is defined as the square root of the sum of the squares of its real and imaginary parts. It plays a fundamental role in characterizing the magnitude of complex numbers and is key when dealing with circles in the complex plane where all points share the same modulus.

Reciprocal of a Complex Number

The reciprocal (or multiplicative inverse) of a nonzero complex number is given by 1/z. Its modulus is the reciprocal of the modulus of z, i.e., |1/z| = 1/|z|. This concept is particularly important when studying complex numbers on circles, such as the unit circle, because it provides symmetry properties that simplify comparisons between a number and its reciprocal.

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