If `ax+x^(2)=y^(2)-ay`, what is a in terms of x and y?

Options

A. `y-x`
B. `x-y`
C. `x+y`
D. `(x^(2)+y^(2))/(x-y)`

Answer

To solve the equation \( ax + x^2 = y^2 - ay \) for \( a \) in terms of \( x \) and \( y \), we can follow these steps:

### Step 1: Rearrange the equation
Start by moving all terms involving \( a \) to one side and the other terms to the opposite side. We can do this by adding \( ay \) to both sides and subtracting \( x^2 \) from both sides:

\[ ax + ay = y^2 - x^2 \]

### Step 2: Factor out \( a \)
Now, we can factor \( a \) out from the left-hand side:

\[ a(x + y) = y^2 - x^2 \]

### Step 3: Use the difference of squares
The right-hand side \( y^2 - x^2 \) can be factored using the difference of squares formula:

\[ y^2 - x^2 = (y - x)(y + x) \]

### Step 4: Substitute back into the equation
Now substitute this back into our equation:

\[ a(x + y) = (y - x)(y + x) \]

### Step 5: Solve for \( a \)
To isolate \( a \), divide both sides by \( (x + y) \):

\[ a = \frac{(y - x)(y + x)}{(x + y)} \]

### Step 6: Simplify if necessary
Since \( (y + x) \) and \( (x + y) \) are the same, we can simplify:

\[ a = y - x \]

### Final Answer
Thus, \( a \) in terms of \( x \) and \( y \) is:

\[ \boxed{y - x} \]

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