Let z̄ denote the complex conjugate of a complex number z. If z is a non-zero complex number for whi

Q: Why did we use polar form $z = r e^{i\theta}$ instead of the standard $z = x + iy$?

Great question! If you use $z = x + iy$, the expression $(\bar{z})^2 + \frac{1}{z^2}$ becomes a nightmare of binomial expansions and complex denominators. Polar form is the 'secret weapon' for problems involving powers and conjugates because it turns multiplication and division into simple addition and subtraction of exponents. It isolates the modulus $r$ from the angle $\theta$.

Q: How do we know $m^2 + n^2$ must be an integer?

The problem explicitly states that the real part ($m$) and the imaginary part ($n$) are integers. Since the set of integers is closed under addition and multiplication, $m^2 + n^2$ must also be an integer. This is the key constraint that allows us to test the options.

Q: Why do we square and add the real and imaginary parts?

We do this to eliminate the $\theta$ dependency. We have $m = A \cos(2\theta)$ and $n = -A \sin(2\theta)$ where $A = r^2 + \frac{1}{r^2}$. Squaring gives $m^2 = A^2 \cos^2(2\theta)$ and $n^2 = A^2 \sin^2(2\theta)$. Adding them yields $m^2 + n^2 = A^2(\cos^2(2\theta) + \sin^2(2\theta)) = A^2$. It's a classic trigonometric identity trick to remove the variable angle.

Question

Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $(\bar{z})^2 + \frac{1}{z^2}$ are integers, then which of the following is/are possible value(s) of $|z|$?

Options:

(A) $\left( \frac{43 + 3\sqrt{205}}{2} \right)^{1/4}$

(B) $\left( \frac{7 + \sqrt{33}}{4} \right)^{1/4}$

(C) $\left( \frac{9 + \sqrt{65}}{4} \right)^{1/4}$

(D) $\left( \frac{7 + \sqrt{13}}{6} \right)^{1/4}$

Sigma Percentile · Accepted Answer

### The Elegance of Polar Symmetry

Welcome, fellow traveler on the path to JEE mastery. Today, we confront a problem that might look like a chaotic mess of complex algebra, but beneath the surface lies a beautiful, symmetric structure.

We are given a non-zero complex number $z$ such that both the real and imaginary parts of $(\bar{z})^2 + \frac{1}{z^2}$ are integers. Our mission is to find the possible value of $|z|$.

### Phase 1: The Polar Transformation

When you see powers like $z^2$ or conjugates like $\bar{z}$ in a complex number problem, your first instinct should be to reach for the polar form. Let $z = r e^{i	heta}$, where $r = |z|$ and $	heta$ is the argument.

The conjugate $\bar{z}$ becomes $r e^{-i	heta}$, and the reciprocal $\frac{1}{z^2}$ becomes $\frac{1}{r^2} e^{-i2	heta}$.

Look at the expression: $(\bar{z})^2 + \frac{1}{z^2}$. Substituting our polar forms, we get:

$$ (r e^{-i	heta})^2 + \frac{1}{r^2 e^{i2	heta}} = r^2 e^{-i2	heta} + \frac{1}{r^2} e^{-i2	heta} $$

Notice the symmetry; both terms share the same exponential factor $e^{-i2	heta}$. We can factor this out:

$$ \left( r^2 + \frac{1}{r^2} ight) e^{-i2	heta} $$

### Phase 2: The Euler Bridge

Now, we invoke Euler's formula: $e^{-i2	heta} = \cos(2	heta) - i \sin(2	heta)$. Our expression becomes:

$$ \left( r^2 + \frac{1}{r^2} ight) (\cos 2	heta - i \sin 2	heta) $$

Let the real part be $m$ and the imaginary part be $n$. Thus:

$$ m = \left( r^2 + \frac{1}{r^2} ight) \cos 2	heta $$

$$ n = -\left( r^2 + \frac{1}{r^2} ight) \sin 2	heta $$

We are told $m$ and $n$ are integers. To find a condition on $r$, we eliminate $	heta$ by squaring and adding:

$$ m^2 + n^2 = \left( r^2 + \frac{1}{r^2} ight)^2 (\cos^2 2	heta + \sin^2 2	heta) $$

Since $\cos^2 2	heta + \sin^2 2	heta = 1$, we are left with the fundamental relation:

$$ m^2 + n^2 = \left( r^2 + \frac{1}{r^2} ight)^2 $$

### Phase 3: The Integer Constraint

Expanding the right side, we get $r^4 + \frac{1}{r^4} + 2 = m^2 + n^2$. Since $m$ and $n$ are integers, $m^2 + n^2$ is an integer.

Therefore, $|z|^4 + \frac{1}{|z|^4} = m^2 + n^2 - 2$ must also be an integer. This serves as our filter to verify potential values of $|z|$.

Let us test the candidate value $|z| = \left( \frac{43 + 3\sqrt{205}}{2} ight)^{1/4}$. Let $R^2 = |z|^4 = \frac{43 + 3\sqrt{205}}{2}$.

Then, the reciprocal is:

$$ \frac{1}{R^2} = \frac{2}{43 + 3\sqrt{205}} $$

Rationalizing by multiplying the numerator and denominator by the conjugate $43 - 3\sqrt{205}$:

$$ \frac{1}{R^2} = \frac{2(43 - 3\sqrt{205})}{43^2 - 9(205)} = \frac{2(43 - 3\sqrt{205})}{1849 - 1845} = \frac{2(43 - 3\sqrt{205})}{4} = \frac{43 - 3\sqrt{205}}{2} $$

Adding $R^2 + \frac{1}{R^2}$ gives:

$$ \frac{43 + 3\sqrt{205} + 43 - 3\sqrt{205}}{2} = \frac{86}{2} = 43 $$

This is an integer. We then check if $43 = m^2 + n^2 - 2$, which implies $m^2 + n^2 = 45$. Since $45 = 6^2 + 3^2$, valid integer solutions for $m$ and $n$ exist.

The logic holds, the math is clean, and the **possible value of $|z|$ is $\left( \frac{43 + 3\sqrt{205}}{2} ight)^{1/4}$**.

Select Answer:

Visualized Solution

$z = r e^{i θ}$

$\overset{z}{ˉ}$ and $\frac{1}{z ^{2}}$

Substitute into Expression

Euler's Formula

The Integer Constraint

Compute $m^{2} + n^{2}$

Expand and Substitute

Testing Option A

Rationalizing $\frac{1}{R ^{2}}$

Checking the Sum

Verifying Integer Solutions

The Sigma Insight: Euler's Form and De Moivre's Theorem

The Elegance of Polar Symmetry

Phase 1

Phase 2

Phase 3

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z=reiθ

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } zˉ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z21​

Substitute into Expression

Euler's Formula

The Integer Constraint

Compute .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m2+n2

Expand and Substitute

Testing Option A

Rationalizing .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R21​

Checking the Sum

Verifying Integer Solutions

The Sigma Insight: Euler's Form and De Moivre's Theorem

The Elegance of Polar Symmetry

Phase 1

Phase 2

Phase 3

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