Let z1 and z2 be any two non-zero complex numbers such that 3|z1| = 4|z2|. If z = 3z1/2z2 + 2z2/3z1

Q: Wait, the problem says $3|z_1| = 4|z_2|$, but that leads to $|w|=2$. Why did you change it to $3|z_1| = 2|z_2|$?

That is a fantastic observation! In many competitive exams, you might encounter a typo in the problem statement. If we use the original $3|z_1| = 4|z_2|$, we get $|w|=2$, which makes $z = 2.5\cos\theta + 1.5i\sin\theta$. This expression is not purely real or imaginary, and none of the options would fit. The problem is clearly designed to test the property that $w + \frac{1}{w}$ is real when $|w|=1$. By correcting the condition to $3|z_1| = 2|z_2|$, the problem becomes mathematically elegant and solvable.

Q: Why is $w + \frac{1}{w}$ always real if $|w|=1$?

Think about the geometry! If $|w|=1$, then $w = \cos\theta + i\sin\theta$. Its reciprocal is $\frac{1}{w} = \bar{w} = \cos\theta - i\sin\theta$. When you add $w$ and $\bar{w}$, the imaginary parts $i\sin\theta$ and $-i\sin\theta$ cancel out perfectly, leaving you with $2\cos\theta$, which is purely real. This is a powerful property of complex numbers on the unit circle.

Q: Can we always assume $|w|=1$ in these types of problems?

Not always, but it is a very common theme in JEE Advanced problems. Whenever you see an expression of the form $w + \frac{1}{w}$, always check the magnitude of $w$. If it is $1$, the expression is real. If it is not $1$, the expression will generally have both real and imaginary parts. Always verify the magnitude first!

Question

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1| = 4|z_2|$. If $z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$ then :

Options:

(A) |z| = \frac{1}{2}\sqrt{\frac{17}{2}}

(B) Re(z) = 0

(C) |z| = \sqrt{\frac{5}{2}}

(D) Im(z) = 0

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the complex plane, looking at the expression:

$$z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$$

At first glance, it looks like a messy algebraic tangle. But as an elite JEE aspirant, you know that complexity is often just a mask for hidden symmetry. Let us peel back that mask together.

### The Power of Substitution

The first step in any complex problem is to simplify the landscape. We are given $3|z_1| = 4|z_2|$ (though we will soon see why this might be a typo).

Let us define a new variable $w = \frac{3z_1}{2z_2}$. Suddenly, the expression for $z$ transforms into something beautiful:

$$z = w + \frac{1}{w}$$

This is the heart of the problem. We are no longer dealing with two separate complex numbers; we are dealing with a number and its reciprocal.

### The Magnitude Trap

Now, let us calculate the magnitude of our new variable $w$. We have:

$$|w| = \left| \frac{3z_1}{2z_2} ight| = \frac{3}{2} \cdot \frac{|z_1|}{|z_2|}$$

Using the given condition, we find the ratio of the moduli. If we follow the path of the intended problem, we realize that for the expression to yield a clean, elegant result, the magnitude of $w$ must be $1$.

This happens if the condition is actually $3|z_1| = 2|z_2|$, leading to:

$$|w| = \frac{3}{2} \cdot \frac{2}{3} = 1$$

This is the **'Aha!' moment**. When $|w|=1$, $w$ lies on the unit circle.

### The Beauty of Euler's Form

When $w$ is on the unit circle, we can express it in its most powerful form: $w = e^{i	heta} = \cos	heta + i\sin	heta$. Consequently, its reciprocal is:

$$\frac{1}{w} = e^{-i	heta} = \cos	heta - i\sin	heta$$

Now, watch what happens when we add them:

$$z = w + \frac{1}{w} = (\cos	heta + i\sin	heta) + (\cos	heta - i\sin	heta)$$

The imaginary parts, $i\sin	heta$ and $-i\sin	heta$, vanish into thin air! We are left with $z = 2\cos	heta$.

### The Final Revelation

Since $z = 2\cos	heta$, it is purely real. It has no imaginary component.

Therefore, **$Im(z) = 0$**. This aligns perfectly with option (4).

The lesson here is not just about solving this specific problem; it is about recognizing the symmetry of complex numbers. Whenever you see $w + \frac{1}{w}$, look for the unit circle. It is the key that unlocks the door to the solution.

Animated Solution for Mathematics - Complex Numbers: Let $z_{1}$ and $z_{2}$ be any two non-zero complex numbers such that $3∣ z_{1} ∣ = 4∣ z_{2} ∣$ . If $z = \frac{3 z _{1}}{2 z _{2}} + \frac{2 z _{2}}{3 z _{1}}$ then :

Select Answer:

Visualized Solution

Analyze the Given Condition

Finding the Ratio of Moduli

Defining a New Variable $w$

Calculating the Magnitude of $w$

Using Polar Form for $w$

Evaluating the Sum $z = w + 1/ w$

The Catch: No Matching Options

Correcting the Condition

Simplified Sum with $∣ w ∣ = 1$

Final Conclusion

The Sigma Insight: Algebraic Operations on Complex Numbers

Analyzing the Setup

The Power of Substitution

The Magnitude Trap

The Beauty of Euler's Form

The Final Revelation

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Select Answer:

Visualized Solution

Analyze the Given Condition

Finding the Ratio of Moduli

Defining a New Variable .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } w

Calculating the Magnitude of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } w

Using Polar Form for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } w

Evaluating the Sum .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z=w+1/w