What is the third proportional to x² − y² and x − y?

Introduction / Context:
This algebraic question generalises the idea of third proportional to symbolic expressions involving x and y. It requires comfort with both the definition of proportionality and algebraic factorisation, particularly the difference of squares identity.

Given Data / Assumptions:

• First term a = x² − y².
• Second term b = x − y.
• We must find the third proportional c such that a : b = b : c.

Concept / Approach:
For numbers a, b and c in proportion a : b = b : c, the relation a / b = b / c holds. This implies a * c = b² and c = b² / a. Here, a is x² − y² and b is x − y. We use the identity x² − y² = (x − y)(x + y) to simplify b² / a and to obtain a simplified expression for the third proportional.

Step-by-Step Solution:
Step 1: Let the third proportional be T. Step 2: By definition, (x² − y²) : (x − y) = (x − y) : T. Step 3: This means (x² − y²) / (x − y) = (x − y) / T. Step 4: Rearranging, (x² − y²) * T = (x − y)². Step 5: Therefore T = (x − y)² / (x² − y²). Step 6: Factorise x² − y² using the difference of squares: x² − y² = (x − y)(x + y). Step 7: Substitute this in the expression for T: T = (x − y)² / [(x − y)(x + y)]. Step 8: Cancel one common factor (x − y) from numerator and denominator, assuming x ≠ y. Step 9: This gives T = (x − y) / (x + y).

Verification / Alternative check:
Check that the proportion holds: (x² − y²) : (x − y) = (x − y) : T. Since x² − y² = (x − y)(x + y), the first ratio becomes [(x − y)(x + y)] : (x − y) which simplifies to (x + y) : 1. The second ratio is (x − y) : [(x − y) / (x + y)] which simplifies, by dividing numerator and denominator by (x − y), to (x + y) : 1 as well. Thus both ratios are equal, verifying that T is correct.

Why Other Options Are Wrong:
Expressions such as x + y or x − y alone do not satisfy the proportionality a : b = b : c for a = x² − y² and b = x − y. Likewise, (x + y) / (x − y) inverts the required expression. Only (x − y) / (x + y) maintains the equality of ratios when the algebraic simplifications are carried out properly.

Common Pitfalls:
Some learners forget the identity x² − y² = (x − y)(x + y) or cancel terms incorrectly. Others mistakenly treat the relation as a three-term arithmetic progression rather than a proportion. It is also easy to invert the final fraction and select (x + y) / (x − y) if the cross multiplication step is rushed. Careful factorisation and cancellation are essential.

Final Answer:
The third proportional to x² − y² and x − y is (x − y) / (x + y).