If |z - 4/z| = 2, then the maximum value of |Z| is equal to (A) √5 + 1 (B) 2 (C) 2 + √2 (D) √3 + 1

Q: Wait, why do we rewrite $z$ as $(z - \frac{4}{z}) + \frac{4}{z}$? Is that even allowed?

Absolutely! This is a classic, elegant technique in complex number problems. We aren't changing the value of $z$; we are just re-labeling it. By adding and subtracting $\frac{4}{z}$, we create a term that matches the expression given in the problem statement ($z - \frac{4}{z}$). This allows us to use the known magnitude of that expression to bound the magnitude of $z$ itself. It's like finding a hidden door in a room you thought was locked.

Q: Why do we ignore the negative root when solving the quadratic inequality $|z|^2 - 2|z| - 4 \le 0$?

That is a fantastic question! Remember that $|z|$ represents the magnitude, or the distance of the complex number $z$ from the origin in the complex plane. By definition, a distance must be a non-negative real number. When we solve the quadratic inequality, we get roots like $1 - \sqrt{5}$ and $1 + \sqrt{5}$. Since $1 - \sqrt{5}$ is negative, it cannot be a valid value for the magnitude $|z|$. Thus, we restrict our focus to the positive range, which is why we only consider the upper bound.

Q: Can we use polar form $z = re^{i\theta}$ to solve this instead?

You certainly can! If you substitute $z = re^{i\theta}$ into the equation $|z - \frac{4}{z}| = 2$, you will get an equation involving $r$ and $\theta$. While this is a valid path, it often leads to more complex trigonometric manipulations. The algebraic approach using the Triangle Inequality is generally faster and less prone to calculation errors for this specific type of problem. Both methods are valid, but always look for the path that minimizes the chance of a slip-up!

Question

If $|z - 4/z| = 2$, then the maximum value of $|Z|$ is equal to

Options:

(A) $\sqrt{5} + 1$

(B) 2$

(C) 2 + \sqrt{2}$

(D) $\sqrt{3} + 1$

Sigma Percentile · Accepted Answer

### The Geometry of Complex Numbers: A Journey of Constraints

Welcome, fellow traveler on the path to JEE mastery! Today, we are going to unravel a problem that might look like a simple algebraic exercise, but is actually a beautiful dance of geometry in the complex plane.

We are given the condition $|z - \frac{4}{z}| = 2$, and our mission is to find the maximum possible value of $|z|$.

### The Art of Manipulation

When you first look at $|z - \frac{4}{z}| = 2$, it feels like a locked box. We have information about the difference between $z$ and its reciprocal, but we want to know about $z$ itself.

The secret lies in a simple, yet profound, algebraic trick. We want to express $z$ in a way that incorporates the term we already know. Let's perform a little magic by adding and subtracting the same term:

$$z = \left(z - \frac{4}{z}\right) + \frac{4}{z}$$

We haven't changed the value of $z$ at all, but we have successfully partitioned it into two parts: one that we know the magnitude of, and one that is just a scaled version of the reciprocal. This is the key to unlocking the problem.

### The Power of the Triangle Inequality

Now that we have $z$ in this form, we need to relate its magnitude to the magnitudes of its parts. This is where the **Triangle Inequality** comes to our rescue.

It states that for any two complex numbers $a$ and $b$, the magnitude of their sum is always less than or equal to the sum of their individual magnitudes:

$$|a + b| \le |a| + |b|$$

Geometrically, this is the statement that the length of one side of a triangle cannot exceed the sum of the lengths of the other two sides. Let's apply this to our expression for $z$, where $a = (z - \frac{4}{z})$ and $b = \frac{4}{z}$:

$$|z| = \left|\left(z - \frac{4}{z}\right) + \frac{4}{z}\right| \le \left|z - \frac{4}{z}\right| + \left|\frac{4}{z}\right|$$

### The Substitution Phase

The pieces of the puzzle now start to fall into place. We know from the problem statement that $|z - \frac{4}{z}| = 2$.

Using the property of moduli that $|\frac{a}{b}| = \frac{|a|}{|b|}$, we know that $|\frac{4}{z}|$ becomes $\frac{4}{|z|}$. Substituting these into our inequality, we get:

$$|z| \le 2 + \frac{4}{|z|}$$

This is a beautiful result! We have successfully turned a complex number problem into a simple inequality involving only the magnitude $|z|$.

### Taming the Quadratic

Since $|z|$ is a magnitude, it is strictly positive, so we can safely multiply the entire inequality by $|z|$ without worrying about flipping the sign. This gives us:

$$|z|^2 \le 2|z| + 4$$

Bringing everything to one side, we get the quadratic inequality:

$$|z|^2 - 2|z| - 4 \le 0$$

Let's treat $|z|$ as a variable, say $x$. We are looking for the range of $x$ that satisfies $x^2 - 2x - 4 \le 0$. To find the boundaries, we solve the equation $x^2 - 2x - 4 = 0$ using the quadratic formula:

$$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-4)}}{2(1)} = \frac{2 \pm \sqrt{20}}{2} = 1 \pm \sqrt{5}$$

### The Final Victory

Our quadratic inequality tells us that $|z|$ must lie between $1 - \sqrt{5}$ and $1 + \sqrt{5}$. Since $|z|$ must be non-negative, the lower bound is effectively zero.

The maximum value, therefore, is the upper bound of our range. The maximum possible value of $|z|$ is **$1 + \sqrt{5}$**.

Animated Solution for Mathematics - Complex Numbers: If $∣ z - 4/ z ∣ = 2$ , then the maximum value of $∣ Z ∣$ is equal to

Select Answer:

Visualized Solution

Understanding the Constraint

The Algebraic Manipulation

The Triangle Inequality

Applying the Inequality

Substituting Known Values

Clearing the Denominator

Forming the Quadratic Inequality

Finding the Critical Points

Calculating the Roots

Analyzing the Range and Maximum

The Sigma Insight: Conjugate and Modulus

The Geometry of Complex Numbers

The Art of Manipulation

The Power of the Triangle Inequality

The Substitution Phase

Taming the Quadratic

The Final Victory

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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−4/z∣=2, then the maximum value of ∣Z∣ is equal to