If \( z = x + iy \), then the equation \( |z + 1| = |z - 1| \) represents:
A circle
A parabola
X-axis
Y-axis
The Correct Option is D
Solution and Explanation
\(|z+1|=|z-1|\)
Let \(z=x+i y\)
Then, \(|x+i y+1|=|x+i y-1|\)
\(\Rightarrow \sqrt{(x+1)^{2}+y^{2}}=\sqrt{(x-1)^{2}+y^{2}}\)
\(\Rightarrow(x+1)^{2}+y^{2}=(x-1)^{2}+y^{2}\)
\(\Rightarrow x^{2}+2 x+1+y^{2}=x^{2}+1-2 x+y^{2}\)
\(\Rightarrow 4 x=0\)
\(\Rightarrow x=0\) which represents \(y-\) axis.
so, The correct option is(D): Y-axis.
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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.