If z ≠ 0 be a complex number such that |z - 1/z| = 2, then the maximum value of |z| is:

Q: Wait, why do we use the reverse triangle inequality here?

The reverse triangle inequality, $||a| - |b|| \leq |a - b|$, is a powerful tool when you have information about the difference of two complex numbers and want to relate it to their moduli. Since we are given $|z - \frac{1}{z}| = 2$, this property allows us to immediately translate that geometric distance into an algebraic inequality involving $|z|$.

Q: Why do we only consider the upper bound $r - \frac{1}{r} \leq 2$?

The question asks for the *maximum* value of $|z|$. The inequality $||r - \frac{1}{r}|| \leq 2$ implies that $r - \frac{1}{r}$ is between $-2$ and $2$. To maximize $r$, we look at the upper limit of this range, which is $r - \frac{1}{r} \leq 2$. The lower bound would give us the minimum, which isn't required here.

Q: Can we just assume $z$ is a real number to solve this?

While it is tempting to assume $z$ is real, $z$ is a complex number. However, the beauty of the triangle inequality is that it holds for all complex numbers. By using the modulus property, we effectively project the problem onto the real line, making the calculation valid for any complex $z$ that satisfies the condition.

Question

If $z 
eq 0$ be a complex number such that $|z - \frac{1}{z}| = 2$, then the maximum value of $|z|$ is:

Options:

(A) $\sqrt{2}$

(B) 1

(C) $\sqrt{2} - 1$

(D) $\sqrt{2} + 1$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given a non-zero complex number $z$ such that the distance between $z$ and its reciprocal, $\frac{1}{z}$, is exactly $2$. Our objective is to determine the maximum possible distance of $z$ from the origin, which is the maximum value of $|z|$.

### The Power of the Triangle Inequality

When dealing with complex numbers and their moduli, the **Reverse Triangle Inequality** is our most trusted tool. It states that for any two complex numbers $a$ and $b$, the absolute difference of their moduli is less than or equal to the modulus of their difference:

$$||a| - |b|| \leq |a - b|$$

This is a fundamental geometric truth, representing the fact that the difference in lengths cannot exceed the length of the vector connecting the two points. By setting $a = z$ and $b = \frac{1}{z}$, we obtain:

$$||z| - |\frac{1}{z}|| \leq |z - \frac{1}{z}|$$

### The Elegant Substitution

We are given that $|z - \frac{1}{z}| = 2$. Recalling that the modulus of a reciprocal is the reciprocal of the modulus, we have $|\frac{1}{z}| = \frac{1}{|z|}$.

Let us define a new variable $r = |z|$, where $r > 0$ as it represents a distance. Substituting these into our inequality, we arrive at a much cleaner expression:

$$|r - \frac{1}{r}| \leq 2$$

This inequality serves as the heart of the problem, constraining the distance $r$. To find the maximum value of $r$, we focus on the upper bound of this absolute value inequality:

$$r - \frac{1}{r} \leq 2$$

### Solving the Quadratic

Since $r$ is a positive distance, we can multiply the entire inequality by $r$ without reversing the inequality sign. This yields:

$$r^2 - 1 \leq 2r$$

Rearranging this into a standard quadratic inequality, we get:

$$r^2 - 2r - 1 \leq 0$$

To find the critical points, we solve the corresponding quadratic equation $r^2 - 2r - 1 = 0$ using the quadratic formula $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

$$r = \frac{2 \pm \sqrt{4 - 4(1)(-1)}}{2} = \frac{2 \pm \sqrt{8}}{2} = 1 \pm \sqrt{2}$$

### The Final Revelation

We have two roots: $1 - \sqrt{2}$ and $1 + \sqrt{2}$. Since the quadratic expression $r^2 - 2r - 1$ must be less than or equal to zero, $r$ must lie between these two roots.

We must respect the physical constraint that $r > 0$. Because $1 - \sqrt{2}$ is negative, the valid range for $r$ is $0 < r \leq 1 + \sqrt{2}$.

Thus, the maximum value of $|z|$ is **$1 + \sqrt{2}$**. This result defines the boundary of the set of all complex numbers satisfying the given condition.

Animated Solution for Mathematics - Complex Numbers: If $z \neq = 0$ be a complex number such that $∣ z - \frac{1}{z} ∣ = 2$ , then the maximum value of $∣ z ∣$ is:

Select Answer:

Visualized Solution

Understanding the Given Condition

Recalling the Triangle Inequality

Applying the Property to $z$ and $\frac{1}{z}$

Substituting the Known Value

Defining $∣ z ∣ = r$

Focusing on the Upper Bound

Forming the Quadratic Inequality

Rearranging to Standard Form

Finding the Roots of the Quadratic

Analyzing the Range of $r$

Conclusion and Final Answer

The Sigma Insight: Conjugate and Modulus

Analyzing the Setup

The Power of the Triangle Inequality

The Elegant Substitution

Solving the Quadratic

The Final Revelation

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List-I

List-II

Animated Solution for Mathematics - Complex Numbers: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z=0 be a complex number such that ∣z−z1​∣=2, then the maximum value of ∣z∣ is: