Problem 60 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(u=\sqrt{a^{2} \cos ^{2} \theta+b^{2} \sin ^{2} \theta}+\sqrt{a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta}\), then the difference between the maximum and minimum values of \(u^{2}\) is given by (A) \(2\left(a^{2}+b^{2}\right)\) (B) \(2 \sqrt{a^{2}+b^{2}}\) (C) \((a+b)^{2}\) (D) \((a-b)^{2}\)

Step-by-Step Solution

Answer

The difference is \((a+b)^2\); option (C) is correct.

Step 1: Simplify the Expression

We start by expanding the expression for \(u\) to find \(u^2\). Since \(u = \sqrt{a^{2} \cos^{2} \theta + b^{2} \sin^{2} \theta} + \sqrt{a^{2} \sin^{2} \theta + b^{2} \cos^{2} \theta}\), we square both sides: \(u^2 = \left(\sqrt{a^{2} \cos^{2} \theta + b^{2} \sin^{2} \theta} + \sqrt{a^{2} \sin^{2} \theta + b^{2} \cos^{2} \theta}\right)^2\). This expands to \(a^{2} + b^{2} + 2\sqrt{(a^{2} \cos^{2} \theta + b^{2} \sin^{2} \theta)(a^{2} \sin^{2} \theta + b^{2} \cos^{2} \theta)}\).

Step 2: Analyze the Squared Terms

Focus on the term inside the square root of the cross product: \((a^{2} \cos^{2} \theta + b^{2} \sin^{2} \theta)(a^{2} \sin^{2} \theta + b^{2} \cos^{2} \theta)\). Compute this to obtain: \(a^{4} + b^{4} - 2a^{2}b^{2}\cos^{2}\theta\sin^{2}\theta\). So the expression for \(u^2\) becomes: \(a^{2} + b^{2} + 2\sqrt{a^{4} + b^{4} - 2a^{2}b^{2}\cos^{2}\theta\sin^{2}\theta}\).

Step 3: Find the Maximum and Minimum Values

From Step 2, the square root term reaches its extrema when \(\cos^{2}\theta\sin^{2}\theta = 0\) or \(\frac{1}{4}\) (because \(\cos^{2}\theta + \sin^{2}\theta = 1\) gives the boundary values), i.e. the least and greatest values for \(2\cos\theta\sin\theta = \sin 2\theta\) range from \(-1\) to \(1\). At \(\theta = 0, \frac{\pi}{2}\) the term is minimal: \(\sqrt{a^2b^2}\). At \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}\) the term is maximal: \(\sqrt{(a^2 - b^2)^2}\).

Step 4: Calculate the Difference

The absolute maximum value of \(u^2\) is \((a + b)^2\) and the minimum is \((a - b)^2\). Therefore, the difference between the maximum and minimum values is \(((a+b)^2 - (a-b)^2 = 4ab\)). This simplifies using algebraic identities.

Step 5: Select the Correct Answer Choice

Given that \(4ab\) corresponds to choice (C), the correct alternative is \((a+b)^2\). Therefore, the answer to the question presented is (C).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. These identities are essential tools in simplifying expressions and proving equations in trigonometry. One of the most commonly used identities is the Pythagorean identity:

• \(\cos^2 \theta + \sin^2 \theta = 1\)

This identity is used extensively when working with trigonometric equations and simplifying expressions, as it helps combine or eliminate terms.

Another useful identity is the double angle identity for sine:

• \(\sin 2\theta = 2\sin\theta\cos\theta\)

This identity helps in simplifying products of sine and cosine in expressions. In the exercise given, understanding the range of values that \(\sin 2\theta\) can take, from \(-1\) to \(1\), allows you to determine the behavior of complex expressions involving trigonometric terms.

Maxima and Minima

Finding the maxima and minima of a function is crucial in mathematics to determine the points where a function achieves its highest or lowest value. The process usually involves taking the derivative of the function, but sometimes, as in this problem, algebraic manipulation combined with trigonometric identities can lead directly to these values.

In the context of the exercise, we deal with maximizing and minimizing the term \(\sqrt{a^4 + b^4 - 2a^2b^2\cos^2\theta\sin^2\theta}\).

• The minimum value is achieved when \( \cos^2\theta\sin^2\theta \) is at zero, which simplifies to \( a^2b^2 \).
• The maximum occurs when \( \cos^2\theta\sin^2\theta = \frac{1}{4} \),maximizing the product, giving \((a^2 - b^2)^2\).

This understanding allows us to compute the difference between these values, showcasing the importance of these concepts in determining critical points in functions involving trigonometric terms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that represent mathematical concepts or quantities. Understanding how to manipulate these expressions is crucial in solving mathematics problems. This manipulation often involves expanding, factoring, or simplifying expressions using fundamental arithmetic properties.

In the problem, the expression for \(u^2\) is initially expanded into its component terms. Such expansions are:

• \(u^2 = a^{2} + b^{2} + 2\sqrt{a^{4} + b^{4} - 2a^{2}b^{2}\cos^2\theta\sin^2\theta}\).

Further simplification might involve the use of trigonometric identities or recognizing perfect squares. The final calculation determining the difference between maxima and minima \((a+b)^2\) and \((a-b)^2\) involves using the identity

• \([x + y]^2 - [x - y]^2 = 4xy.\)

Understanding these algebraic manipulations is key to solving such problems efficiently.