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
The correct answer is/are option(s):
(C): zero
(D): purely imaginary
\(|\frac{z_1+z_2}{z_1-z_2}|=1\)
Now,
\(\Rightarrow |z_1+z_2|=|z_1-z_2|\)
\(\Rightarrow |z_1+z_2|^2=|z_1-z_2|^2\)
\(\Rightarrow (z_1+z_2)(z_1+z_2)=(z_1-z_2)(z_1-z_2)\)
\(\Rightarrow|z_1|^2+|z_2|^2+z_1z_2+z_1z_2=|z_1|^2+|z_2|^2-z_1z_2-z_1z_2\)
\(\Rightarrow 2(z_1z_2+z_2z_1)=0\)
\(\Rightarrow \frac{z_1}{z_2}+\frac{z_1}{z_2}=0\)
\(\Rightarrow 2 Re (\frac{z_1}{z_2})=0=\frac{z_1}{z_2}\)
Hence it is purely imazinary.
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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.