The third proportional to `(x^(2) - y^(2)) and (x-y)` is :

Options

A. `(x+y)`
B. `(x-y)`
C. `(x+y)/(x-y)`
D. `(x-y)/(x+y)`

Answer

To find the third proportional to \( (x^2 - y^2) \) and \( (x - y) \), we follow these steps:

### Step 1: Understand the concept of third proportional
The third proportional to two numbers \( a \) and \( b \) is given by the formula:
\[ \text{Third Proportional} = \frac{b^2}{a} \]
In our case, \( a = x^2 - y^2 \) and \( b = x - y \).

### Step 2: Apply the formula
Using the formula for the third proportional, we substitute \( a \) and \( b \):
\[ \text{Third Proportional} = \frac{(x - y)^2}{(x^2 - y^2)} \]

### Step 3: Simplify the expression
Now, we simplify \( x^2 - y^2 \) using the difference of squares:
\[ x^2 - y^2 = (x - y)(x + y) \]
Thus, we can rewrite our expression:
\[ \text{Third Proportional} = \frac{(x - y)^2}{(x - y)(x + y)} \]

### Step 4: Cancel common terms
Since \( (x - y) \) is common in the numerator and the denominator, we can cancel it out:
\[ \text{Third Proportional} = \frac{x - y}{x + y} \]

### Final Answer
The third proportional to \( (x^2 - y^2) \) and \( (x - y) \) is:
\[ \frac{x - y}{x + y} \]

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