Question

The harmonic mean of {(a)/(1-ab)\ and {(a)/(1+ab)} is equal to :}

• A. (a/b)
• B. a
• C. (a)/(1-ab)
• D. 1/a

Hint:

When computing the harmonic mean of algebraic fractions, it is often simpler to compute the sum of their reciprocals first before dividing 2 by the result.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The harmonic mean is a type of numerical average, typically used when finding the average of rates or fractions.
Key Formula or Approach:
For any two non-zero real numbers \( x \) and \( y \), their harmonic mean (\(\text{HM}\)) is defined as:
\[ \text{HM} = \frac{2}{\frac{1}{x} + \frac{1}{y}} = \frac{2xy}{x+y} \]

Step 2: Detailed Explanation:
Let us designate the two expressions as \( x \) and \( y \):
\[ x = \frac{a}{1 - ab} \]
\[ y = \frac{a}{1 + ab} \]
We first compute the reciprocals of both terms:
\[ \frac{1}{x} = \frac{1 - ab}{a} \]
\[ \frac{1}{y} = \frac{1 + ab}{a} \]
Now, we find the sum of these reciprocals:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1 - ab}{a} + \frac{1 + ab}{a} \]
Combining the terms over the common denominator \( a \):
\[ \frac{1}{x} + \frac{1}{y} = \frac{(1 - ab) + (1 + ab)}{a} = \frac{2}{a} \]
Substituting this sum back into the harmonic mean formula:
\[ \text{HM} = \frac{2}{\frac{1}{x} + \frac{1}{y}} = \frac{2}{\frac{2}{a}} = a \]
Thus, the harmonic mean simplifies to \( a \).

Step 3: Final Answer:
The harmonic mean is equal to a.