Problem 57 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z_{1}\) and \(z_{2}\) are the two complex roots of equal magnitude and their arguments differ by \(\frac{\pi}{2}\), of the quadratic equation \(a x^{2}+b x+c=0(a \neq 0)\) then \(a\) (in terms of \(b\) and \(c\) ) is (A) \(\frac{b^{2}}{2 c}\) (B) \(\frac{b^{2}}{c}\) (C) \(\frac{b}{2 c}\) (D) None of these

Step-by-Step Solution

Answer

The answer is (A) \(\frac{b^2}{2c}\).

Step 1: Understand the Problem

We need to find the value of \(a\) in terms of \(b\) and \(c\) when the quadratic equation \(ax^2 + bx + c = 0\) has complex roots \(z_1\) and \(z_2\) of equal magnitude, and their arguments differ by \(\frac{\pi}{2}\). The condition on their arguments implies a specific geometric relationship.

Step 2: Roots with Equal Magnitude

Since both roots have equal magnitude, suppose \(z_1 = r\text{cis}(\theta)\) and \(z_2 = r\text{cis}(\theta + \frac{\pi}{2})\), where \(\text{cis}(\theta) = \cos(\theta) + i\sin(\theta)\). The roots are \(r\text{cis}(\theta)\) and \(r\text{cis}(\theta + \frac{\pi}{2}) = ir\text{cis}(\theta)\).

Step 3: Roots as Quadratic Equation Solutions

For the quadratic equation, the sum of the roots \(z_1 + z_2\) is \(-\frac{b}{a}\), and the product \(z_1 z_2 = \frac{c}{a}\). From the expressions for \(z_1\) and \(z_2\), we have \(z_1 + z_2 = r(1 + i)\) and \(z_1 z_2 = r^2i\).

Step 4: Solve for Coefficients

Equate to coefficients: \(r(1+i) = -\frac{b}{a}\) and \(r^2 i = \frac{c}{a}\). From \(r^2 i = \frac{c}{a}\), solving gives \(r^2 = -\frac{c}{a}i^{-1} = \frac{ci}{a}\), considering \(i^{-1} = -i\).

Step 5: Eliminate Variables to Find \(a\)

Using \(r^2 = \frac{c}{a}(-i)\), and since \(r=-\frac{b}{a(1+i)}\), equate \(r^2\) to get \(\left(-\frac{b}{a(1+i)}\right)^2 = \frac{c}{a}(-i)\). Expand and simplify to solve for \(a\).

Step 6: Solve for \(a\) in Terms of \(b\) and \(c\)

Solving \(\left(-\frac{b}{a(1+i)}\right)^2 = \frac{c}{a}(-i)\), get \(\frac{b^2}{2ai} = \frac{c}{a}(-i)\) which simplifies to \(b^2 = 2ac\) and so \(a = \frac{b^2}{2c}\).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically in the form of \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). Quadratics are fundamental in algebra and appear in various mathematical and real-world applications. They can model the trajectory of a projectile, financial forecasts, or even physical phenomena, like motion under constant acceleration.

Solving a quadratic equation involves finding the values of \(x\) (also known as roots), which satisfy the equation. The most common methods to solve quadratics include:

• Factoring: Expressing the quadratic in terms of its factors.
• Quadratic Formula: Using the formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\) to find the roots.
• Completing the Square: Rewriting the equation in a perfect square form.
• Graphing: Using the graph of the equation to find the points where it crosses the x-axis.

Each method has its advantages, depending on the specific equation and context. In some cases, especially when complex numbers are involved, using the quadratic formula becomes essential.

Roots of Equations

The roots of an equation are the values that satisfy the equation, setting it to zero. For quadratic equations, roots may either be real or complex numbers.

Real roots can be found when the discriminant (\(b^2 - 4ac\)) is non-negative. When the discriminant is greater than zero, there are two distinct real roots. If it equals zero, the quadratic has one real root, known as a repeated or double root.

Complex roots appear when the discriminant is negative. In such cases, the quadratic has two complex conjugate roots, arising from the square root of a negative number. However, in some scenarios, like the given exercise, the complex roots can have specific properties. Here, the roots have equal magnitude and arguments that differ by \(\frac{\pi}{2}\). These roots lie on the complex plane at 90 degrees to each other, signifying a special geometric relationship.

Trigonometric Form of Complex Numbers

Complex numbers can be expressed in trigonometric form, providing a clear picture of their position and behavior in the complex plane. A complex number \(z\) can be written as \(z = r \text{cis}(\theta)\), where:

• \(r\) is the magnitude or modulus of the number.
• \(\theta\) is the argument, the angle formed with the positive real axis.
• \(\text{cis}(\theta) = \cos(\theta) + i\sin(\theta)\).

This form is particularly useful in multiplying and dividing complex numbers. The product of two complex numbers involves multiplying their magnitudes and adding their arguments, while division requires dividing their magnitudes and subtracting their arguments.

In the context of the exercise, the roots \(z_1\) and \(z_2\) are expressed in trigonometric form to highlight their equal magnitude and differing arguments. They can be written as \(z_1 = r \text{cis}(\theta)\) and \(z_2 = r \text{cis}(\theta + \frac{\pi}{2})\). This representation is crucial for solving the problem as it translates a geometric relationship into an algebraic one that can be operated on to find \(a\) in terms of \(b\) and \(c\).