Problem 96 - 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 correct answer is (A) \( \frac{b^2}{2c} \).

Step 1 - Recognize the conditions

We are given that the roots of the quadratic equation are complex and have equal magnitudes. This implies the roots can be represented as \( z_1 = r e^{i\theta} \) and \( z_2 = r e^{i(\theta + \frac{\pi}{2})} \), where \( \theta \) is the argument of one root, and \( r \) is their common magnitude.

Step 2 - Relate the roots to the coefficients

For a quadratic equation \( ax^2 + bx + c = 0 \), the sum and product of roots (\( z_1 \) and \( z_2 \)) are given by the relations \( z_1 + z_2 = -\frac{b}{a} \) and \( z_1 z_2 = \frac{c}{a} \).

Step 3 - Calculate sum of roots

Using the polar form of roots, \( z_1 + z_2 = r e^{i\theta} + r e^{i(\theta + \frac{\pi}{2})} = r (e^{i\theta} + i e^{i\theta}) = r e^{i\theta}(1 + i) \).

Step 4 - Calculate product of roots

The product of the roots \( z_1 z_2 = r e^{i\theta} \times r e^{i(\theta + \frac{\pi}{2})} = r^2 e^{i(2\theta + \frac{\pi}{2})} \). Since \( z_1 z_2 = \frac{c}{a} \), this implies \( r^2 = \frac{c}{a} \).

Step 5 - Express in terms of real numbers

From Step 3, the magnitude of \( z_1 + z_2 = r \sqrt{2} e^{i\theta} \) implies \( \left| -\frac{b}{a} \right| = r \sqrt{2} \). Hence, \( \frac{b}{a} = \sqrt{2} r e^{i\alpha} \) where \( e^{i\alpha} \) is some complex unit.

Step 6 - Relate the modulus

As \( b/a = \sqrt{2} r \), taking modulus gives \( \left| \frac{b}{a} \right| = \sqrt{2} r \). Since \( r^2 = \frac{c}{a} \), it follows that \( r = \sqrt{\frac{c}{a}} \).

Step 7 - Express \( a \) in terms of knowns

From Step 6, substituting \( r \) back, \( \sqrt{\frac{2c}{a}} = \frac{b}{a} \). Rearrange for \( a \) to obtain \( a = \frac{b^2}{2c} \).

Step 8 - Verify solution

The obtained expression for \( a \) matches option (A) \( \frac{b^2}{2c} \).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These equations can have either real or complex roots. The solutions or roots of a quadratic equation can be found using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Depending on the value of the discriminant \( b^2 - 4ac \):

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, there is a single real root (a repeated root).
• If the discriminant is negative, the equation has two complex roots.

Understanding these conditions is crucial in solving and analyzing the nature of the roots of a quadratic equation, especially when dealing with complex numbers.

Complex Roots

Complex roots arise when the discriminant of a quadratic equation is negative. These roots come in conjugate pairs and are expressed as \( a + bi \) and \( a - bi \), where \( i \) is the imaginary unit such that \( i^2 = -1 \). In the context of our given exercise, the roots are expressed in polar form as \( z_1 = r e^{i\theta} \) and \( z_2 = r e^{i(\theta + \frac{\pi}{2})} \), indicating that they are complex conjugates with equal magnitude but differing arguments.

• The polar form of a complex number is particularly useful because it makes multiplication and division much simpler, by transforming complex arithmetic into operations on magnitudes and arguments.
• The condition that their arguments differ by \( \frac{\pi}{2} \) means they are perpendicular in the complex plane.

This specific setup can be exploited to better understand the relationships between the coefficients of the quadratic equation and its roots, as seen in this exercise.

Magnitude of Complex Numbers

The magnitude (or modulus) of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \). It represents the distance of the point \((a, b)\) from the origin in the complex plane. In polar form, a complex number is represented as \( z = r e^{i\theta} \), where \( r \) is the magnitude and \( \theta \) is the argument (or angle) of the complex number.

• Magnitude is an essential aspect when solving quadratic equations with complex roots, as seen in calculating \( r^2 = \frac{c}{a} \).
• Using the magnitude can simplify expressions and relationships, like equating \( |\frac{b}{a}| = \sqrt{2} r \) with real coefficients.
• This concept is particularly highlighted in the exercise where roots have the same magnitude, leading to specific algebraic manipulations to find the desired coefficients.

Understanding the magnitude helps in navigating through the solutions of quadratic equations with complex numbers, bridging between different representations and their implications.