If z1 and z2 are two complex numbers satisfying the equation\(|\frac{z_1+z_2}{z_1-z_2}|=1\), then \(\frac{z_1}{z_2}\) may be
- real positive
- real negative
- zero
- purely imaginary
The Correct Option is C, D
Solution and Explanation
Given: \[ \left| \frac{z_1 + z_2}{z_1 - z_2} \right| = 1 \Rightarrow |z_1 + z_2| = |z_1 - z_2| \] Now squaring both sides: \[ |z_1 + z_2|^2 = |z_1 - z_2|^2 \] Use identity: \[ (z_1 + z_2)(\overline{z_1 + z_2}) = (z_1 - z_2)(\overline{z_1 - z_2}) \Rightarrow |z_1|^2 + |z_2|^2 + z_1 \overline{z_2} + \overline{z_1} z_2 = |z_1|^2 + |z_2|^2 - z_1 \overline{z_2} - \overline{z_1} z_2 \] Subtracting both sides: \[ 2(z_1 \overline{z_2} + \overline{z_1} z_2) = 0 \Rightarrow 2 \text{Re} \left( \frac{z_1}{z_2} \right) = 0 \Rightarrow \text{Re} \left( \frac{z_1}{z_2} \right) = 0 \] Therefore, \( \frac{z_1}{z_2} \text{ is purely imaginary} \Rightarrow \text{Correct options: }\) (C) zero, (D) purely imaginary
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.