If (2a+3b) (2c -3d) = (2a-3b) (2c+3d) then

Options

A. `(a)/(b) = (c )/(d) `
B. `(a)/(d) = ( c)/(b) `
C. `(a)/(b) = (d)/(c )`
D. `(b)/(a) = (c )/(d)`

Answer

To solve the equation \((2a + 3b)(2c - 3d) = (2a - 3b)(2c + 3d)\), we will follow these steps:

### Step 1: Expand both sides of the equation
We will expand both sides of the equation using the distributive property (also known as the FOIL method for binomials).

**Left Side:**
\[ (2a + 3b)(2c - 3d) = 2a \cdot 2c + 2a \cdot (-3d) + 3b \cdot 2c + 3b \cdot (-3d) \]
Calculating each term:
\[ = 4ac - 6ad + 6bc - 9bd \]

**Right Side:**
\[ (2a - 3b)(2c + 3d) = 2a \cdot 2c + 2a \cdot 3d + (-3b) \cdot 2c + (-3b) \cdot 3d \]
Calculating each term:
\[ = 4ac + 6ad - 6bc - 9bd \]

### Step 2: Set the expanded forms equal to each other
Now we have the equation:
\[ 4ac - 6ad + 6bc - 9bd = 4ac + 6ad - 6bc - 9bd \]

### Step 3: Simplify the equation
Subtract \(4ac\) and \(-9bd\) from both sides:
\[ -6ad + 6bc = 6ad - 6bc \]
Now, we can rearrange the equation:
\[ -6ad - 6ad = -6bc - 6bc \]
This simplifies to:
\[ -12ad = -12bc \]

### Step 4: Divide both sides by -12
Dividing both sides by -12 gives:
\[ ad = bc \]

### Step 5: Rearranging the equation
Now, we can rearrange this to find the ratio:
\[ \frac{a}{b} = \frac{c}{d} \]

### Conclusion
Thus, we conclude that:
\[ \frac{a}{b} = \frac{c}{d} \]