Problem 62 - A Complete Resource Book in Mathematics for JEE Main

Question

If the quadratic equation \(z^{2}+(a+i b) z+c+i d=0\), where \(a, b, c, d\) are non-7ero real numbers, has a real root, then (A) \(d^{2}-a b d-c^{2}=0\) (B) \(d^{2}-a b d+b^{2} c=0\) (C) \(d^{2}+a b d+c^{2}=0\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (B) \(d^{2}-a b d+b^{2} c=0\).

Step 1: Identify the condition for real roots

For the quadratic equation \( z^2 + (a + ib)z + (c + id) = 0 \) to have a real root, the imaginary part must be zero. Let the real root be \( z = r \). Then \( z = r \) satisfies the equation.

Step 2: Substitute the real root into the equation

Substitute \( z = r \) into the equation: \[ r^2 + (a + ib)r + c + id = 0 \]Separating real and imaginary parts, we get:\[ r^2 + ar + c = 0 \] (Real Part) and \[ br + d = 0 \] (Imaginary Part)

Step 3: Solve for r in the imaginary equation

From the imaginary equation \( br + d = 0 \), solve for \( r \): \[ r = -\frac{d}{b} \] Ensure \( b eq 0 \) since it is given as non-zero.

Step 4: Substitute r back into the real part equation

Substitute \( r = -\frac{d}{b} \) into the real part equation:\[ \left(-\frac{d}{b}\right)^2 + a\left(-\frac{d}{b}\right) + c = 0 \]Solve the equation:\[ \frac{d^2}{b^2} - \frac{ad}{b} + c = 0 \]Multiply through by \( b^2 \) to clear fractions:\[ d^2 - abd + b^2c = 0 \]

Step 5: Match with provided options

From the previous step, the resulting equation is:\[ d^2 - abd + b^2c = 0 \]Compare this with the options provided. Option (B) matches.

Key Concepts

Real Roots

A quadratic equation can possess real roots, and this occurs when there is no imaginary component to the solution. In simpler terms, a real root means that the solution or solutions to the equation are real numbers, which can be plotted on the real number line. Consider the quadratic equation in standard form as in the problem: \[ z^2 + (a + ib)z + (c + id) = 0 \]where \(a, b, c,\) and \(d\) are non-zero real numbers. For this specific equation to have a real root, the imaginary component must equal zero. To achieve this:

• Identify any terms associated with the imaginary unit \(i\).
• Set the equation's imaginary part to zero, which simplifies the problem to only handle real numbers.

Recognizing real roots hinges on understanding these concepts, thus paving the way to solving similar problems effectively.

Complex Numbers

Complex numbers combine real numbers and imaginary numbers, forming the complex plane where any number can be expressed as \(a + ib\). In this expression, \(a\) is the real part and \(ib\) the imaginary part, with \(i\) being the imaginary unit satisfying \(i^2 = -1\).In the quadratic equation from our problem, the terms \((a + ib)z\) and \( (c + id) \) introduce complexity as each element could potentially introduce imaginary parts into the equation. Whenever addressing equations with complex coefficients:

• Separate the real and imaginary terms clearly to allow individual simplification.
• Focus on the relationship between these components to identify any real or complex roots.

Complex numbers extend our ability to solve equations beyond what is possible using only real numbers, providing a comprehensive understanding of different kinds of solutions.

Imaginary Part

The imaginary part of a complex number is crucial in determining whether certain quadratic equations can have real roots. For a quadratic equation like \(z^2 + (a + ib)z + (c + id) = 0\), isolating and considering the imaginary part provides insight into potential solutions.In our process, the imaginary part is derived from substituting the assumed root into the quadratic equation:\[ br + d = 0 \]This equation results directly from setting the imaginary component to zero, a necessary condition for ensuring real roots. Here are the steps to solve for the imaginary part correctly:

• Isolate the terms involving \(i\) and equate them to zero.
• Solve the resulting expression to find constraints on the variable \(r\), which represent potential real roots.

Mastering this concept helps demystify the role of imaginary parts in equations and ensures full comprehension of real and complex solutions.