Problem 194 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(z_{1}\) and \(z_{2}\) be two roots of the equation \(z^{2}+a z+b=0\), \(z\) being complex. Further, assume that the origin, \(z_{1}\) and \(z_{2}\) form an equilateral triangle, then \([2003]\) (A) \(a^{2}=b\) (B) \(a^{2}=2 b\) (C) \(a^{2}=3 b\) (D) \(a^{2}=4 b\)

Step-by-Step Solution

Answer

The correct answer is (C) \(a^2 = 3b\).

Step 1: Understand Vieta's formulas

Given the quadratic equation \(z^2 + az + b = 0\), by Vieta's formulas, the sum of the roots \(z_1 + z_2 = -a\) and the product of the roots \(z_1 z_2 = b\).

Step 2: Triangle properties

Since \(z_1\), \(z_2\), and the origin form an equilateral triangle, the distance between each pair of these points must be equal. This means \(|z_1| = |z_2| = |z_1 - z_2|\).

Step 3: Explore magnitudes

The equality \(|z_1 - z_2| = |z_1| = |z_2|\) implies that the distance between \(z_1\) and \(z_2\) is the same as their distance from the origin. Thus, \(|z_1 - z_2|^2 = |z_1|^2 = |z_2|^2\).

Step 4: Use the distance formula

We have \(|z_1|^2 = z_1 \overline{z_1}\) and \(|z_2|^2 = z_2 \overline{z_2}\), and \(|z_1 - z_2|^2 = (z_1 - z_2)(\overline{z_1} - \overline{z_2})\). Known that \(|z_1 - z_2|^2 = |z_1|^2\), we get \((z_1 - z_2)(\overline{z_1} - \overline{z_2}) = z_1 \overline{z_1}\)."

Step 5: Relationship with coefficients

Substitute from Vieta's formulas where \(z_1 + z_2 = -a\), we know \(|z_1 - z_2| = \sqrt{(z_1 - z_2)(\overline{z_1} - \overline{z_2})}\). This simplifies to \((z_1 - z_2)^2 = 3(z_1)(z_2)\).

Step 6: Solve for \(a^2\) in terms of \(b\)

Using \((z_1 - z_2)^2 = |z_1|^2\) and substituting with \(z_1 = -\frac{a}{2} + \frac{\sqrt{3}(b^{1/2})}{2}\), previously noted that \(|z_1 - z_2|^2 = 3b\), it becomes \(a^2 = 3b\). The solution aligns with option \((C)\) \(a^2 = 3b\).

Key Concepts

Vieta's Formulas

Vieta's formulas are a set of equalities relating the coefficients of a polynomial to sums and products of its roots. They are particularly useful in understanding quadratic equations. For a quadratic equation like \( z^2 + az + b = 0 \), Vieta's formulas tell us two important things:

• The sum of the roots \( z_1 + z_2 = -a \)
• The product of the roots \( z_1 z_2 = b \)

Thanks to Vieta's formulas, we know a lot about the roots without needing to actually calculate them directly. Instead, we can derive relationships and create equations that help solve complex problems, such as determining the geometric properties when these roots form an equilateral triangle.

Equilateral Triangle Properties

An equilateral triangle has three sides that are all equal in length. When it comes to complex numbers forming an equilateral triangle, certain distance relationships must hold. In the context of roots of quadratic equations, if \( z_1 \) and \( z_2 \) are the roots and they, along with the origin, form an equilateral triangle, we observe the following:

• The distances \(|z_1|\), \(|z_2|\), and \(|z_1 - z_2|\) must be identical.

This equality reflects that each side of the triangle, which in this scenario translates to each complex distance, is equal. This property provides us with some important equations we can use: \(|z_1| = |z_2| = |z_1 - z_2|\). By leveraging this condition, we can explore deeper relationships between the roots and the coefficients of the polynomial.

Quadratic Equations

Quadratic equations are expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients. Solving quadratic equations typically involves finding the values of \( x \) that satisfy the equation. For equations with complex coefficients or roots, such as \( z^2 + az + b = 0 \), the methods remain similar.

One significant aspect is that these roots can represent points in the complex plane, giving rise to geometric interpretations. In our exercise, the relationship \((z_1 - z_2)^2 = 3(z_1z_2)\) highlights the interaction between algebraic roots and geometric triangles. Therefore, understanding how the coefficients relate to root properties illuminates the complex geometry of quadratic equations, offering more than just numeric solutions but a full geometric context of the problem at hand.