Let z1, z2 be the roots of the equation z² + az + 12 = 0 and z1, z2 form an equilateral triangle wit

Q: Why can't we just use the distance formula? It seems easier.

The distance formula involves square roots and moduli, which often leads to messy, non-algebraic expressions. By using the complex number condition for an equilateral triangle, we stay within the realm of polynomial algebra, which is much cleaner and less prone to calculation errors. In JEE Advanced, elegance is often the shortest path to the correct answer.

Q: Wait, why does $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$ work? Is there a derivation?

This identity is derived from the rotation property. If you rotate the vector $(z_1 - z_3)$ by $60^\circ$ to get $(z_2 - z_3)$, you get $(z_2 - z_3) = (z_1 - z_3)e^{i\pi/3}$. Squaring and rearranging this leads directly to that beautiful symmetric identity. It is a standard tool in your JEE arsenal that saves you from doing trigonometry in the complex plane!

Q: Do I need to worry about the sign of $a$?

The question asks for $|a|$, the absolute value. Since our calculation leads to $a^2 = 36$, we have $a = 6$ or $a = -6$. In both cases, the magnitude $|a|$ is $6$. Always check what the question is asking for—don't lose points on a sign error!

Question

Let $z_1, z_2$ be the roots of the equation $z^2 + az + 12 = 0$ and $z_1, z_2$ form an equilateral triangle with origin. Then, the value of $|a|$ is

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer. Today, we are not just solving a quadratic equation; we are embarking on a journey through the Argand plane. When you look at a problem involving complex numbers, I want you to stop seeing just variables and coefficients. I want you to see points, vectors, and shapes.

This problem is a classic example of how the rigid, structured world of algebra meets the fluid, beautiful world of geometry.

### The Algebraic DNA

We start with the quadratic equation $z^2 + az + 12 = 0$. In the JEE Advanced landscape, whenever you see a quadratic equation, your first instinct should be to invoke the spirit of Vieta. Vieta's formulas are the DNA of the polynomial; they tell us everything we need to know about the roots without ever having to solve for them explicitly.

We know that for any quadratic $z^2 + az + b = 0$, the sum of the roots is $-a$ and the product of the roots is $b$. Here, our roots are $z_1$ and $z_2$. Thus, we immediately extract two vital pieces of information:

1. The sum: $z_1 + z_2 = -a$

2. The product: $z_1 z_2 = 12$

Keep these tucked away in your mental toolkit. They are the bridge we will cross later.

### The Geometric Insight

Now, let's shift our gaze to the Argand plane. We are told that $z_1$, $z_2$, and the origin ($0$) form an equilateral triangle. This is a powerful geometric constraint. In the complex plane, an equilateral triangle is not just any shape; it is a manifestation of symmetry.

There is a standard, elegant condition for three complex numbers $z_1, z_2, z_3$ to form an equilateral triangle:

$$z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$$

Why does this work? Think of it as a rotation. To get from one vertex to another in an equilateral triangle, you are essentially rotating the vector by $60^{\circ}$ (or $\pi/3$ radians). When you translate this rotation into the language of complex numbers using the operator $e^{i\pi/3}$, this identity emerges as the ultimate simplification.

### The Simplification

Here is where the problem rewards your intuition. We are told one of the vertices is the origin. This means $z_3 = 0$. Let's look at our massive identity again. If we plug in $z_3 = 0$, the terms containing $z_3$ simply vanish into thin air:

$$z_1^2 + z_2^2 + (0)^2 = z_1z_2 + z_2(0) + (0)z_1$$

Suddenly, the equation collapses into something incredibly manageable:

$$z_1^2 + z_2^2 = z_1z_2$$

This is the moment of clarity. We have successfully reduced a complex geometric condition into a simple algebraic relationship.

### The Final Synthesis

We are almost there. We have $z_1^2 + z_2^2 = z_1z_2$, but our Vieta's formulas gave us $(z_1 + z_2)$ and $(z_1 z_2)$. We need to bridge this gap. We use the classic algebraic identity for the sum of squares:

$$z_1^2 + z_2^2 = (z_1 + z_2)^2 - 2z_1z_2$$

Substituting this into our simplified condition, we get:

$$(z_1 + z_2)^2 - 2z_1z_2 = z_1z_2$$

Rearranging the terms, we find:

$$(z_1 + z_2)^2 = 3z_1z_2$$

This is the 'Aha!' moment. We have linked the geometry of the triangle directly to the coefficients of our quadratic equation. Now, we simply substitute our values from the algebraic phase:

$$(-a)^2 = 3(12)$$

$$a^2 = 36$$

### Conclusion

Finally, we take the square root. The question asks for $|a|$, the absolute value of $a$. Since $a^2 = 36$, we conclude that **$|a| = 6$**.

Look at what you just did. You didn't just calculate a number; you navigated the relationship between the roots of a polynomial and the symmetry of the Argand plane. You used Vieta's formulas to capture the essence of the roots, and you used the equilateral condition to capture the essence of the geometry. That is the heart of JEE Advanced mathematics—seeing the connections where others see only separate formulas.

Animated Solution for Mathematics - Complex Numbers: Let $z_{1}, z_{2}$ be the roots of the equation $z^{2} + a z + 12 = 0$ and $z_{1}, z_{2}$ form an equilateral triangle with origin. Then, the value of $∣ a ∣$ is

Enter Numerical Value:

Visualized Solution

Visualizing the Problem

Sum and Product of Roots

Equilateral Triangle Condition

Simplifying with Origin

Algebraic Transformation

Final Equation Setup

Substitution and Calculation

The Final Answer

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Setup

The Algebraic DNA

The Geometric Insight

The Simplification

The Final Synthesis

Conclusion

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