Area of the parallelogram formed by the lines
`a_(1)x+b_(1)y+c_(1)=0,a_(1)x+b_(1)y+d_(1)=0`
and `a_(2)x+b_(2)y+c_(2)=0,a_(2)x+b_(2)y+d_(2)=0` is
`([d_(1)-c_(1))(d_(2)-c_(2)))/([(a_(1)^(2)+b_(1)^(2))(a_(2)^(2)+b_(2)^(2))]^(1//2))`

Answer

To find the area of the parallelogram formed by the lines given in the question, we will follow these steps:

### Step 1: Understand the equations of the lines
The lines given are:
1. \( a_1x + b_1y + c_1 = 0 \)
2. \( a_1x + b_1y + d_1 = 0 \)
3. \( a_2x + b_2y + c_2 = 0 \)
4. \( a_2x + b_2y + d_2 = 0 \)

These lines can be rewritten in slope-intercept form if needed, but for the area calculation, we will use their coefficients directly.

### Step 2: Identify the distance between the parallel lines
The distance \( d \) between two parallel lines of the form \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by the formula:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
For our lines, the distance between the first pair of lines is:
\[ d_1 = \frac{|d_1 - c_1|}{\sqrt{a_1^2 + b_1^2}} \]
And for the second pair of lines:
\[ d_2 = \frac{|d_2 - c_2|}{\sqrt{a_2^2 + b_2^2}} \]

### Step 3: Calculate the area of the parallelogram
The area \( A \) of the parallelogram formed by these two pairs of parallel lines can be calculated using the formula:
\[ A = d_1 \times d_2 \]
Substituting the distances we found:
\[ A = \left(\frac{|d_1 - c_1|}{\sqrt{a_1^2 + b_1^2}}\right) \times \left(\frac{|d_2 - c_2|}{\sqrt{a_2^2 + b_2^2}}\right) \]

### Step 4: Combine the results
Thus, the area of the parallelogram can be expressed as:
\[ A = \frac{|d_1 - c_1| \cdot |d_2 - c_2|}{\sqrt{(a_1^2 + b_1^2)(a_2^2 + b_2^2)}} \]
This matches the given formula in the question:
\[ A = \frac{(d_1 - c_1)(d_2 - c_2)}{\sqrt{(a_1^2 + b_1^2)(a_2^2 + b_2^2)}} \]

### Final Result
The area of the parallelogram formed by the given lines is:
\[ A = \frac{(d_1 - c_1)(d_2 - c_2)}{\sqrt{(a_1^2 + b_1^2)(a_2^2 + b_2^2)}} \]