Problem 9 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\sin \theta=\frac{1}{2}\left(\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}\right)\) necessarily implies (A) \(x>y\) (B) \(x

Step-by-Step Solution

Answer

(C) \(x=y\)

Step 1: Understand the Expression

We have an equation involving trigonometric and algebraic expressions: \(\sin \theta = \frac{1}{2}\left(\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}}\right)\).

Step 2: Apply Trigonometric Constraint

Recognize that \(\sin \theta \) ranges from -1 to 1. The maximum possible value of \(\frac{1}{2}\left(\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}}\right)\) therefore must also be within this range.

Step 3: Simplify Symmetric Terms

Notice \(\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}}\) achieves a maximum value when \(x=y\) since this would yield \(2\), which matches the limit imposed by \(\sin \theta \).

Step 4: Conclusion

Since the sum of \(\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}}\) reaches its maximum value of 2 when \(x = y\), this is the value that satisfies the constraint for the sin function.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In our exercise, the expression \( \sin \theta = \frac{1}{2}\left(\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}}\right) \) is an example of an algebraic expression that also involves variables and radicals. The key to understanding such expressions is to break them down into simpler parts. For instance, in our expression, two terms \( \sqrt{\frac{x}{y}} \) and \( \sqrt{\frac{y}{x}} \) are combined. These terms involve operations that include square roots and are involved in an addition operation, all scaled by \( \frac{1}{2} \).

• The square root operation implies taking the square root of a division of variables.
• The 'plus' symbol signifies addition of the two square root operations.
• The term \( \frac{1}{2} \) means that the sum of the operations is halved.

Understanding algebraic expressions with radicals require familiarity with basic algebraic operations and adherence to the order of operations to correctly simplify and evaluate the expression.

Sin Function

The sine function, written as \( \sin \theta \), is one of the fundamental trigonometric functions. It relates the angle \( \theta \) of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. In our exercise, the sine function is set equal to a particular algebraic expression. It's important to remember that the sine function always has values between -1 and 1, inclusive.

• When dealing with equations involving \( \sin \theta \), ensure that your solution respects this range.
• This property constrains other elements in the equation, like our algebraic expression, to also fit within the -1 to 1 range.

Understanding the behavior of the sine function, especially the range constraint, is crucial for solving trigonometric equations. In this specific case, recognizing that any transformation of the equation also fits in this range ensures that our solution is possible.

Equations Involving Radicals

Equations involving radicals, such as square roots, often require careful manipulation to isolate variable terms and solve. In the exercise, we encounter radicals in the expression \( \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} \). A powerful insight when dealing with such expressions is recognizing symmetry.

• A perfect balance or symmetry can simplify the radicals dramatically.
• In this problem, symmetry arises when \( x = y \), resulting in equal terms within the equation.
• When simplified under this condition, the expression inside the sine function peaks at its maximum value, perfectly matching the possible maximum value of \( \sin \theta \), which is 1.

The understanding of symmetry and the behavior of radicals under these conditions is vital. Recognizing when variables are equal helps simplify complex expressions, streamline calculations, and reach correct conclusions more efficiently.