What will be the area of the rhombus `ax pm by pm c = 0` ?

Options

A. `(3c^(2))//(ab)`
B. `(4c^(2))//(ab)`
C. `(2c^(2))//(ab)`
D. `(c^(2))//(ab)`

Answer

To find the area of the rhombus represented by the equation \( ax + by + c = 0 \), we can follow these steps:

### Step 1: Identify the equations of the lines
The rhombus is formed by the following four lines:
1. \( ax + by + c = 0 \)
2. \( ax + by - c = 0 \)
3. \( ax - by + c = 0 \)
4. \( ax - by - c = 0 \)

### Step 2: Find the points of intersection
We need to find the intersection points of these lines to determine the vertices of the rhombus.

- For the first line \( ax + by + c = 0 \):
- Set \( x = 0 \) to find the y-intercept:
\[ y = -\frac{c}{b} \quad \text{(Point: } (0, -\frac{c}{b})\text{)} \]
- Set \( y = 0 \) to find the x-intercept:
\[ x = -\frac{c}{a} \quad \text{(Point: } (-\frac{c}{a}, 0)\text{)} \]

- For the second line \( ax + by - c = 0 \):
- Set \( x = 0 \):
\[ y = \frac{c}{b} \quad \text{(Point: } (0, \frac{c}{b})\text{)} \]
- Set \( y = 0 \):
\[ x = \frac{c}{a} \quad \text{(Point: } (\frac{c}{a}, 0)\text{)} \]

- For the third line \( ax - by + c = 0 \):
- Set \( x = 0 \):
\[ y = \frac{c}{b} \quad \text{(Point: } (0, \frac{c}{b})\text{)} \]
- Set \( y = 0 \):
\[ x = -\frac{c}{a} \quad \text{(Point: } (-\frac{c}{a}, 0)\text{)} \]

- For the fourth line \( ax - by - c = 0 \):
- Set \( x = 0 \):
\[ y = -\frac{c}{b} \quad \text{(Point: } (0, -\frac{c}{b})\text{)} \]
- Set \( y = 0 \):
\[ x = \frac{c}{a} \quad \text{(Point: } (\frac{c}{a}, 0)\text{)} \]

### Step 3: Identify the vertices of the rhombus
From the points calculated, we have the vertices of the rhombus as:
1. \( (0, \frac{c}{b}) \)
2. \( (0, -\frac{c}{b}) \)
3. \( (\frac{c}{a}, 0) \)
4. \( (-\frac{c}{a}, 0) \)

### Step 4: Calculate the lengths of the diagonals
The lengths of the diagonals \( D_1 \) and \( D_2 \) can be calculated as follows:
- \( D_1 \) (vertical diagonal) is the distance between \( (0, \frac{c}{b}) \) and \( (0, -\frac{c}{b}) \):
\[ D_1 = \left| \frac{c}{b} - \left(-\frac{c}{b}\right) \right| = \frac{2c}{b} \]

- \( D_2 \) (horizontal diagonal) is the distance between \( (\frac{c}{a}, 0) \) and \( (-\frac{c}{a}, 0) \):
\[ D_2 = \left| \frac{c}{a} - \left(-\frac{c}{a}\right) \right| = \frac{2c}{a} \]

### Step 5: Calculate the area of the rhombus
The area \( A \) of the rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times D_1 \times D_2 \]
Substituting the values of \( D_1 \) and \( D_2 \):
\[ A = \frac{1}{2} \times \left(\frac{2c}{b}\right) \times \left(\frac{2c}{a}\right) = \frac{2c^2}{ab} \]

### Final Answer
Thus, the area of the rhombus is:
\[ \text{Area} = \frac{2c^2}{ab} \]