To find the equation of the diagonal of the square formed by the pairs of lines given by the equations \(xy + 4x - 3y - 12 = 0\) and \(xy - 3x + 4y - 12 = 0\), we can follow these steps:
### Step 1: Rearranging the First Equation
Start with the first equation:
\[
xy + 4x - 3y - 12 = 0
\]
Rearranging gives:
\[
xy - 3y + 4x - 12 = 0
\]
This can be factored or rearranged to isolate terms involving \(x\) and \(y\).
### Step 2: Finding Intersections from the First Equation
From the rearranged form, we can express \(y\) in terms of \(x\):
\[
y(x - 3) = 12 - 4x
\]
Thus, we can find specific values of \(x\) and \(y\) by substituting values or solving for intersections.
### Step 3: Rearranging the Second Equation
Now consider the second equation:
\[
xy - 3x + 4y - 12 = 0
\]
Rearranging gives:
\[
xy + 4y - 3x - 12 = 0
\]
Similar to the first equation, we can isolate \(y\) or \(x\).
### Step 4: Finding Intersections from the Second Equation
From the rearranged form, express \(y\) in terms of \(x\):
\[
y(x + 4) = 3x + 12
\]
Again, we can find specific values of \(x\) and \(y\) by substituting values or solving for intersections.
### Step 5: Identifying Intersection Points
By solving both equations, we can find the points of intersection. For example, if we set \(x = 3\) and \(y = -4\) from the first equation, and \(x = -4\) and \(y = 3\) from the second equation, we can determine the coordinates of the vertices of the square formed by these lines.
### Step 6: Finding the Diagonal Equation
The diagonals of the square can be determined by the coordinates of the points found. If we label the points as \(A(-4, -4)\), \(B(-4, 3)\), \(C(3, 3)\), and \(D(3, -4)\), we can find the equations of the diagonals \(AC\) and \(BD\).
1. For diagonal \(AC\):
- The slope \(m\) can be calculated using the coordinates of points \(A\) and \(C\).
- The equation can be written in the form \(y - y_1 = m(x - x_1)\).
2. For diagonal \(BD\):
- Similarly, calculate the slope using points \(B\) and \(D\) and derive the equation.
### Step 7: Final Equation of the Diagonal
After calculating, we find that the equation of the diagonal can be expressed in standard form.
The final equations derived from the calculations yield:
1. For diagonal \(AC\): \(y = x\)
2. For diagonal \(BD\): \(x + y + 1 = 0\)
### Final Answer
Thus, the equation of the diagonal of the square formed by the pairs of lines is:
\[
x + y + 1 = 0
\]
