Problem 94 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the arithmetic mean of two numbers a and \(\mathrm{b}, \mathrm{a}>\mathrm{b}>0\), is five times their geometric mean, then \(\frac{\mathrm{a}+\mathrm{b}}{\mathrm{a}-\mathrm{b}}\) is equal to : [Online April 8, 2017| (a) \(\frac{\sqrt{6}}{2}\) (b) \(\frac{3 \sqrt{2}}{4}\) (c) \(\frac{7 \sqrt{3}}{12}\) (d) \(\frac{5 \sqrt{6}}{12}\)

Step-by-Step Solution

Answer

Option (d): \(\frac{5\sqrt{6}}{12}\) is correct.

Step 1: Understand the Relationship between Means

The arithmetic mean of two numbers \(a\) and \(b\) is given by \(\frac{a+b}{2}\), while the geometric mean is \(\sqrt{ab}\). According to the problem, the arithmetic mean is five times the geometric mean. Thus, we can set up the equation: \[ \frac{a+b}{2} = 5\sqrt{ab}. \]

Step 2: Clear Fractions and Square Both Sides

First, we'll clear the fraction by multiplying both sides of the equation by 2: \[ a + b = 10\sqrt{ab}. \] To eliminate the square root, square both sides: \[ (a+b)^2 = (10\sqrt{ab})^2. \] This simplifies to: \[ a^2 + 2ab + b^2 = 100ab. \]

Step 3: Rearrange and Simplify the Equation

Rearrange the equation to bring like terms together: \[ a^2 + b^2 = 100ab - 2ab. \] This simplifies to: \[ a^2 + b^2 = 98ab. \]

Step 4: Express \(a^2 + b^2\) in Terms of \((a-b)^2\)

Recall the identity \(a^2 + b^2 = (a-b)^2 + 2ab\). Substitute this into our simplified equation: \[ (a-b)^2 + 2ab = 98ab. \] This simplifies to \[ (a-b)^2 = 96ab. \]

Step 5: Compute \(\frac{a+b}{a-b}\) Using Relations Found

We know from step 1 that \(a+b = 10\sqrt{ab}\) and from step 4, \((a-b)^2 = 96ab\), hence \(a-b = \sqrt{96ab}\). Simplifying \(\sqrt{96} = 4\sqrt{6}\), so \(a-b = 4\sqrt{6}\sqrt{ab}\). Therefore, \[ \frac{a+b}{a-b} = \frac{10\sqrt{ab}}{4\sqrt{6}\sqrt{ab}} = \frac{10}{4\sqrt{6}} = \frac{5}{2\sqrt{6}}. \] Multiply numerator and denominator by \(\sqrt{6}\) to rationalize: \[ \frac{5\sqrt{6}}{12}. \]

Step 6: Conclusion

Thus, \(\frac{a+b}{a-b} = \frac{5\sqrt{6}}{12}\). This matches option (d).

Key Concepts

Geometric Mean

The geometric mean between two numbers is a concept derived from the field of mathematics, offering a different type of average than the more familiar arithmetic mean. It is represented as the square root of the product of two numbers. Mathematically, if you have numbers \(a\) and \(b\), the geometric mean is \(\sqrt{ab}\). This average is particularly useful when the numbers involved are products of values, and it is often used in growth rates and financial contexts. Compared to the arithmetic mean, which adds values and divides by the count, the geometric mean assesses multiplicative processes, providing a measure that aptly reflects the central tendency of exponential or proportional data. By understanding both types of means, you gain a comprehensive toolset for analyzing numerical data in different contexts.
One main characteristic is that the geometric mean is always less than or equal to the arithmetic mean, which has applications in the inequalities often seen in mathematical problems.

Rationalizing Fractions

Rationalizing fractions involves removing radicals from the denominator of a fraction. This process is important because it simplifies the fraction, making it easier to work with in mathematical operations and analyses. When you encounter a denominator with a radical, for example \(\frac{5}{2\sqrt{6}}\), the goal is to eliminate the square root from the denominator.
To rationalize, you multiply both the numerator and the denominator by the radical found in the denominator. In our example, you multiply by \(\sqrt{6}\), resulting in \(\frac{5\sqrt{6}}{2\cdot6}\), which simplifies to \(\frac{5\sqrt{6}}{12}\).
This technique is not just an exercise in simplification but a foundational skill required for algebraic manipulation across various scientific and mathematical fields, ensuring that calculations and interpretations are both accurate and manageable. This method also paves the way to a better understanding of operations concerning irrational numbers.

Equations

Equations form the backbone of algebra and are a critical tool for solving problems in mathematics. An equation is a mathematical statement that asserts the equality of two expressions. It usually involves one or more variables. For instance, if you have the equation \(a + b = 10\sqrt{ab}\), it shows a relationship where the sum of \(a\) and \(b\) equals ten times their geometric mean.
Solving equations often requires manipulating these expressions to isolate the variables. This step-by-step process can involve various techniques, such as moving terms across the equals sign by adding, subtracting, multiplying, or dividing, and using tools like substitution or elimination. Equations may also require solving quadratic expressions, which often happens when dealing with squared terms, as seen in the original problem that necessitated simplifying \((a+b)^2 = 100ab\).
When working with equations, each step should maintain the equation's balance to find the correct solution, serving as the key to unlocking unknown quantities in diverse mathematical contexts.