Problem 82 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z=a+i b\) where \(a>0, b>0\), then (A) \(|z| \geq \frac{1}{\sqrt{2}}(a-b)\) (B) \(|z| \geq \frac{1}{\sqrt{2}}(a+b)\) (C) \(|z|<\frac{1}{\sqrt{2}}(a+b)\) (D) None of these

Step-by-Step Solution

Answer

Correct answer is (A): \(|z| \geq \frac{1}{\sqrt{2}}(a-b)\).

Step 1: Understand the Complex Number

We have a complex number given as \( z = a + ib \), where \( a > 0 \) and \( b > 0 \). This means that \( a \) is the real part and \( b \) is the imaginary part of the complex number \( z \).

Step 2: Compute the Magnitude of the Complex Number

The magnitude of the complex number \( z = a + ib \) is calculated using the formula: \[ |z| = \sqrt{a^2 + b^2} \]

Step 3: Analyze the Given Options

We need to compare the magnitude \( |z| = \sqrt{a^2 + b^2} \) with each of the given inequalities:- (A) \(|z| \geq \frac{1}{\sqrt{2}}(a-b)\) - (B) \(|z| \geq \frac{1}{\sqrt{2}}(a+b)\)- (C) \(|z| < \frac{1}{\sqrt{2}}(a+b)\)We analyze each by testing if \( |z| \) meets the given inequality.

Step 4: Test Option (A)

For option (A), \(|z| \geq \frac{1}{\sqrt{2}}(a-b)\):- Recall \( |z| = \sqrt{a^2 + b^2} \)- The inequality becomes \( \sqrt{a^2 + b^2} \geq \frac{1}{\sqrt{2}}(a-b) \)- Square both sides: \(a^2 + b^2 \geq \frac{1}{2}(a^2 - 2ab + b^2)\)- This simplifies to \( a^2 + b^2 \geq a^2 - 2ab + b^2 \), which is always true since \( 2ab \geq 0 \).- Therefore, option (A) holds true.

Step 5: Test Option (B)

For option (B), \(|z| \geq \frac{1}{\sqrt{2}}(a+b)\):- The inequality is \( \sqrt{a^2 + b^2} \geq \frac{1}{\sqrt{2}}(a+b) \)- Squaring both sides gives \( a^2 + b^2 \geq \frac{1}{2}(a^2 + 2ab + b^2) \)- Simplifying results in \( a^2 + b^2 \geq a^2 + 2ab + b^2 \), which is not true because \( 2ab \) is positive.- Thus, option (B) does not hold true.

Step 6: Test Option (C)

For option (C), \(|z| < \frac{1}{\sqrt{2}}(a+b)\):- We have \( \sqrt{a^2 + b^2} < \frac{1}{\sqrt{2}}(a+b) \)- Squaring both sides gives \( a^2 + b^2 < \frac{1}{2}(a^2 + 2ab + b^2) \)- After simplification, this contradicts the inequality \( a^2 + b^2 < a^2 + 2ab + b^2 \).- Therefore, option (C) cannot be true either.

Step 7: Conclusion

Based on the analysis, option (A) is the only correct statement among the given options.

Key Concepts

Magnitude of Complex Numbers

The concept of the magnitude of complex numbers is essential for understanding the distance of a complex number from the origin in the complex plane. You can think of a complex number as a point on a two-dimensional plane with a real and an imaginary axis. The number is represented as \( z = a + ib \), where \( a \) and \( b \) are real numbers indicating the components on these axes.To find the magnitude or modulus of this complex number, we use the formula:\[|z| = \sqrt{a^2 + b^2}\]This formula is akin to the Pythagorean Theorem and shows that the magnitude is the length of the vector from the origin to the point \( (a, b) \) in this plane.

• \( |z| \) represents the distance from the origin to the point \( (a, b) \)
• It is always a non-negative real number

Understanding this concept is crucial for many operations and comparisons involving complex numbers.

Complex Number Inequalities

Inequalities involving complex numbers often require the comparison of their magnitudes rather than individual components. Given a complex number \( z = a + ib \), we are sometimes asked to look at how its magnitude compares to other expressions.In exercises like the one provided, we're tasked with testing the truth of inequalities such as:

• \(|z| \geq \frac{1}{\sqrt{2}}(a-b)\)
• \(|z| \geq \frac{1}{\sqrt{2}}(a+b)\)
• \(|z| < \frac{1}{\sqrt{2}}(a+b)\)

To determine if these inequalities hold true, we replace \(|z|\) with \(\sqrt{a^2 + b^2}\) and simplify both sides of the inequality. It's crucial to note that the multiplication by \(\frac{1}{\sqrt{2}}\) scales the right side of the inequalities, providing us with different scenarios to test and understand.

Real and Imaginary Components

Complex numbers have two components: the real part and the imaginary part. Each part plays a specific role in the properties of the complex number.The real component is represented by \( a \) in \( z = a + ib \). It indicates the position of the number along the real (or horizontal) axis of the complex plane. Similarly, \( b \) is the imaginary component and corresponds to its position along the imaginary (or vertical) axis.

• Real Part (\(a\)): Horizontal position on the complex plane
• Imaginary Part (\(b\)): Vertical position on the complex plane

Both components define the overall direction and location of the complex number on the plane. Positive values of \(a\) and \(b\) indicate the number resides in the first quadrant of the complex plane, offering insights into more complex evaluations like inequalities and geometric interpretations.