Question:
The number of complex numbers $z$ such that $|z - 1| = |z + 1| = |z - i|$ equals
The number of complex numbers $z$ such that $|z - 1| = |z + 1| = |z - i|$ equals
Updated On: Aug 1, 2022
- $1$
- $2$
- $\infty$
- $0$
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The Correct Option is A
Solution and Explanation
Let $z = x + iy$
$|z - 1| = |z + 1|\quad\Rightarrow Re \,z = 0 \quad\Rightarrow x = 0$
$|z - 1| = |z - i|\quad\Rightarrow x = y$
$|z + 1| = |z - i|\quad\Rightarrow y = -x$
Only $\left(0, 0\right)$ will satisfy all conditions.
$\Rightarrow$ Number of complex number $z = 1$
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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.