Animated Solution for Mathematics - Complex Numbers: Let $Z_1$ and $Z_2$ be two roots of the equation $Z^2+aZ+b=0$, $Z$ being complex. Further, assume that the origin, $Z_1$ and $Z_2$ form an equilateral triangle. Then

Imagine you are standing on the complex plane, looking at the origin $O(0)$ and two complex numbers $Z_1$ and $Z_2$. These three points form a perfectly symmetric equilateral triangle.

This geometric constraint implies rotational symmetry. If you rotate $Z_1$ about the origin by $60^{\circ }$, you land on $Z_2$. This symmetry is the key to unlocking the relationship between the coefficients of the quadratic equation $Z^2+aZ+b=0$.

Vieta's Formulas

We are given the quadratic equation $Z^2+aZ+b=0$. From the fundamental theorem of algebra, we know that the roots $Z_1$ and $Z_2$ are intimately connected to the coefficients $a$ and $b$.

Specifically, the sum of the roots is given by:

The product of the roots is given by:

These two relations serve as our bridge between the abstract world of complex roots and the concrete world of coefficients.

A Powerful Tool

For any three complex numbers $Z_1,Z_2,Z_3$ to form an equilateral triangle, they must satisfy the following condition:

In our specific case, one of the vertices is the origin, so we set $Z_3=0$. When we substitute $Z_3=0$ into this condition, the terms $Z_3^2$, $Z_2{Z}_3$, and $Z_3{Z}_1$ vanish.

We are left with the simplified constraint:

We connect the geometric constraint $Z_1^2+Z_2^2=Z_1{Z}_2$ with our algebraic relations using the identity:

Substituting this into our constraint, we obtain:

Rearranging the terms leads to:

Substituting the Vieta's relations $(-a{)}^2=3(b)$, we arrive at the final condition:

$$a^2=3b$$

We started with a geometric shape and, through the power of complex algebra, arrived at a clean, elegant condition. This is the beauty of mathematics, where the visual and the algebraic are woven into a single truth.

You have successfully navigated this classic JEE problem. Remember: every complex problem is just a simple truth waiting to be uncovered.