What is the fourth proportional of `x ^(2) -y ^(2) , x + y , x -y.`

Options

A. 0
B. 2
C. 1
D. 4

Answer

To find the fourth proportional of the numbers \( x^2 - y^2 \), \( x + y \), and \( x - y \), we can follow these steps:

### Step 1: Understand the concept of fourth proportional
The fourth proportional \( P \) to three numbers \( a, b, c \) is defined such that the ratio \( \frac{a}{b} = \frac{c}{P} \).

### Step 2: Set up the equation
In our case, we have:
- \( a = x^2 - y^2 \)
- \( b = x + y \)
- \( c = x - y \)

Thus, we can write the equation:
\[ \frac{x^2 - y^2}{x + y} = \frac{x - y}{P} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ (x^2 - y^2) \cdot P = (x + y) \cdot (x - y) \]

### Step 4: Simplify the right-hand side
The right-hand side can be simplified using the difference of squares:
\[ (x + y)(x - y) = x^2 - y^2 \]
So the equation now looks like:
\[ (x^2 - y^2) \cdot P = x^2 - y^2 \]

### Step 5: Solve for \( P \)
Now, we can divide both sides by \( x^2 - y^2 \) (assuming \( x^2 - y^2 \neq 0 \)):
\[ P = 1 \]

### Conclusion
Thus, the fourth proportional of \( x^2 - y^2 \), \( x + y \), and \( x - y \) is:
\[ \boxed{1} \]

---