If `ax^(2)-bx=ay^(2)+by`, then `(a)/(b)=`

Options

A. `(1)/(x-y)`
B. `(1)/(x+y)`
C. `(x-y)/(x+y)`
D. `(x+y)/(x-y)`

Answer

To solve the equation \( ax^2 - bx = ay^2 + by \) and find the value of \( \frac{a}{b} \), we can follow these steps:

### Step 1: Rearrange the equation
Start by moving all terms involving \( a \) to one side and all terms involving \( b \) to the other side:

\[ ax^2 - ay^2 = bx + by \]

### Step 2: Factor out common terms
Now, factor out \( a \) from the left side and \( b \) from the right side:

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

### Step 3: Divide both sides
Next, divide both sides by \( b(x + y) \) (assuming \( b \neq 0 \) and \( x + y \neq 0 \)):

\[ \frac{a}{b} = \frac{x + y}{x^2 - y^2} \]

### Step 4: Simplify the right side
Notice that \( x^2 - y^2 \) can be factored as \( (x + y)(x - y) \):

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

### Step 5: Cancel common terms
Now, we can cancel \( x + y \) from the numerator and denominator:

\[ \frac{a}{b} = \frac{1}{x - y} \]

### Final Answer
Thus, the value of \( \frac{a}{b} \) is:

\[ \frac{a}{b} = \frac{1}{x - y} \]

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