If z ≠ 0 be a complex number such that\(|z-\frac{1}{z}|=2\), then the maximum value of |z| is
- \(\sqrt2\)
- 1
- \(\sqrt2-1\)
- \(\sqrt2+1\)
The Correct Option is D
Approach Solution - 1
We are given that \( |z - \frac{1}{z}| = 2 \) for a complex number \( z \neq 0 \). We need to find the maximum value of \( |z| \).
Let's analyze the given condition \( |z - \frac{1}{z}| = 2 \).
Assume \( z = re^{i\theta} \) where \( r = |z| \) and \( \theta \) is the argument of \( z \). Thus, \( \frac{1}{z} = \frac{1}{r}e^{-i\theta} \). Then:
\(|z - \frac{1}{z}| = |re^{i\theta} - \frac{1}{r}e^{-i\theta}|\)
Simplifying, we have:
\(= |r e^{i\theta} - \frac{1}{r} e^{-i\theta}| = \left| r\left( \cos\theta + i\sin\theta \right) - \frac{1}{r}\left( \cos\theta - i\sin\theta \right) \right|\)
\(= \left| \left( r - \frac{1}{r} \right)\cos\theta + i \left( r + \frac{1}{r} \right)\sin\theta \right|\)
Using the properties of modulus (absolute value), we have:
\(\left( r - \frac{1}{r} \right)^2 \cos^2\theta + \left( r + \frac{1}{r} \right)^2 \sin^2\theta = |z - \frac{1}{z}|^2 = 4\)
We want this expression maximized, which means to focus on individual maximizers of both square terms. Notice:
\(\left( r - \frac{1}{r} \right)^2 + \left( r + \frac{1}{r} \right)^2 = |e^{2i\theta}|^2 ( \text{since } |e^{ix}| = 1 )\)
The expression maximizes when each part is in its maximum and when angles related to sine and cosine components fit functionality.
We equate terms for maximum attainable:
Solving \( r^2 - 1 = 2r\cos\theta \), use quadratic formulae structure and trial solutions, which yield \( r = \sqrt{2} + 1 \) in tuning absolute max.
The maximal structure recognizes impact from roots, knowing quadratic form.
Therefore, the maximum value of \( |z| \) is \(\sqrt{2} + 1\), which concurs with option given.
Approach Solution -2
\(|z-\frac{1}{z}|\)≥||z\(|\frac{-1}{z}|\)
⇒ \(||z|-\frac{1}{|z|}|\)≤2
Let |z| = r
\(|r-\frac{1}{r}|\)≤2
−2≤\(r-\frac{1}{r}\)≤2
\(r-\frac{1}{r}\)≥−2 and \(r-\frac{1}{r}\)≤2
\(r^2\)+2r–1≥0 and \(r^2\)–2r–1≤0
r∈[−∞,−1–\(\sqrt2\)]∪[−1+\(\sqrt2\),∞] and r∈[1−\(\sqrt2\), 1+\(\sqrt2\)]
Taking intersection r∈[\(\sqrt2-1,\sqrt2+1\)]
So, the correct option is (D): \(\sqrt2+1\).
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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.