If `((a-b)^(2)-(a+b)^(2))/(-4a)= (x)/(y)`, on simplifying the above mentioned equation, what will be the equation?

Options

A. `xy= b`
B. bx= y
C. by= x
D. ab=x

Answer

To solve the equation \(\frac{(a-b)^2 - (a+b)^2}{-4a} = \frac{x}{y}\), we will simplify the left side step by step.

### Step 1: Expand the squares
We start by expanding \((a-b)^2\) and \((a+b)^2\):
\[ (a-b)^2 = a^2 - 2ab + b^2 \]
\[ (a+b)^2 = a^2 + 2ab + b^2 \]

### Step 2: Substitute the expansions into the equation
Substituting these expansions into the equation gives us:
\[ \frac{(a^2 - 2ab + b^2) - (a^2 + 2ab + b^2)}{-4a} = \frac{x}{y} \]

### Step 3: Simplify the numerator
Now, simplify the numerator:
\[ (a^2 - 2ab + b^2) - (a^2 + 2ab + b^2) = a^2 - 2ab + b^2 - a^2 - 2ab - b^2 \]
This simplifies to:
\[ -4ab \]

### Step 4: Substitute back into the equation
Now, we substitute back into our equation:
\[ \frac{-4ab}{-4a} = \frac{x}{y} \]

### Step 5: Simplify the fraction
The \(-4\) cancels out, and we can simplify:
\[ \frac{ab}{a} = \frac{x}{y} \]
This further simplifies to:
\[ b = \frac{x}{y} \]

### Final Result
Thus, the simplified equation is:
\[ b = \frac{x}{y} \]

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