Understanding the Third Proportional Concept

The question asks for the third proportional to two given terms: \((x^2 - y^2)\) and \((x - y)\). Let the two given terms be 'a' and 'b', and the third proportional be 'c'. The concept of third proportional states that the ratio of the first term to the second term is equal to the ratio of the second term to the third term. This can be written as a proportion:

\(\frac{a}{b} = \frac{b}{c}\)

To find the third proportional 'c', we can rearrange this equation:

\(c = \frac{b \times b}{a} = \frac{b^2}{a}\)

Applying the Formula to Find the Third Proportional

In this question, we have:

• First term, \(a = (x^2 - y^2)\)
• Second term, \(b = (x - y)\)

We need to find the third proportional, \(c\). Using the formula \(c = \frac{b^2}{a}\), we substitute the given terms:

\(c = \frac{(x - y)^2}{(x^2 - y^2)}\)

Simplifying the Expression for the Third Proportional

To simplify this expression, we can use the algebraic identity for the difference of squares, which states that \(x^2 - y^2 = (x - y)(x + y)\).

Substitute this into the denominator of our expression for \(c\):

\(c = \frac{(x - y)^2}{(x - y)(x + y)}\)

We can expand the term \((x - y)^2\) in the numerator as \((x - y)(x - y)\):

\(c = \frac{(x - y)(x - y)}{(x - y)(x + y)}\)

Assuming that \((x - y) \neq 0\) (i.e., \(x \neq y\)), we can cancel out one factor of \((x - y)\) from both the numerator and the denominator:

\(c = \frac{x - y}{x + y}\)

This simplified expression is the third proportional to \((x^2 - y^2)\) and \((x - y)\).

Verifying the Third Proportional with Options

Let's compare our result, \(\frac{x - y}{x + y}\), with the given options:

• Option 1: \((x - y)\) - Does not match.
• Option 2: \(\rm \frac{x-y}{x+y}\) - Matches our result.
• Option 3: \(\rm \frac{x+y}{x-y}\) - Does not match.
• Option 4: \((x + y)\) - Does not match.

Therefore, the third proportional is \(\frac{x - y}{x + y}\).

Revision Table: Key Concepts

Additional Information on Proportionality

Proportionality is a fundamental concept in mathematics used to describe relationships between quantities. Besides the third proportional, other related concepts include:

• Fourth Proportional: For three numbers a, b, and c, the fourth proportional d is such that \(\frac{a}{b} = \frac{c}{d}\). Here, \(d = \frac{bc}{a}\).
• Mean Proportional: For two numbers a and c, the mean proportional b is such that \(\frac{a}{b} = \frac{b}{c}\). Here, \(b^2 = ac\), so \(b = \sqrt{ac}\). The mean proportional is also sometimes called the geometric mean.

Understanding these concepts helps in solving various problems involving ratios and proportions in algebra, geometry, and other areas of mathematics.