The diagonals of the parallelogram whose sides are `lx+my+n = 0`,`lx+ my+ n',=0`, `mx+ly+n=0`, `mx+ly+n'=0` include an angle

Options

A. `pi//3`
B. `pi//2`
C. `tan^(-1)((l^(2)-m^(2))/(l^(2)+m^(2)))`
D. `tan^(-1)((2lm)/(l^(2)+m^(2)))`

Answer

To find the angle between the diagonals of the parallelogram formed by the given equations, we will follow these steps:

### Step 1: Identify the equations of the sides of the parallelogram
The sides of the parallelogram are given by the following equations:
1. \( lx + my + n = 0 \) (Equation 1)
2. \( lx + my + n' = 0 \) (Equation 2)
3. \( mx + ly + n = 0 \) (Equation 3)
4. \( mx + ly + n' = 0 \) (Equation 4)

### Step 2: Determine the intersection points of the lines
To find the vertices of the parallelogram, we need to find the intersection points of the pairs of lines.

#### Finding point A (intersection of lines 1 and 3):
1. Solve the equations \( lx + my + n = 0 \) and \( mx + ly + n = 0 \).
2. Rearranging gives us:
- From Equation 1: \( my = -lx - n \)
- Substitute \( y \) in Equation 3:
\[ mx + l\left(-\frac{lx + n}{m}\right) + n = 0 \]
3. Solve for \( x \) and then substitute back to find \( y \).

#### Finding point B (intersection of lines 1 and 4):
1. Solve \( lx + my + n = 0 \) and \( mx + ly + n' = 0 \).
2. Follow similar steps as for point A.

#### Finding point C (intersection of lines 2 and 3):
1. Solve \( lx + my + n' = 0 \) and \( mx + ly + n = 0 \).

#### Finding point D (intersection of lines 2 and 4):
1. Solve \( lx + my + n' = 0 \) and \( mx + ly + n' = 0 \).

### Step 3: Calculate the coordinates of the vertices
After solving the equations, we will get the coordinates for points A, B, C, and D.

### Step 4: Find the slopes of the diagonals
The diagonals of the parallelogram are AC and BD.

1. **Slope of AC**:
- Let the coordinates of A be \( (x_A, y_A) \) and C be \( (x_C, y_C) \).
- The slope \( m_1 \) is given by:
\[ m_1 = \frac{y_C - y_A}{x_C - x_A} \]

2. **Slope of BD**:
- Let the coordinates of B be \( (x_B, y_B) \) and D be \( (x_D, y_D) \).
- The slope \( m_2 \) is given by:
\[ m_2 = \frac{y_D - y_B}{x_D - x_B} \]

### Step 5: Find the angle between the diagonals
To find the angle \( \theta \) between the two diagonals, we use the formula:
\[ \tan(\theta) = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \]

### Step 6: Determine if the diagonals are perpendicular
If \( m_1 \cdot m_2 = -1 \), then the diagonals are perpendicular, and the angle \( \theta \) is \( 90^\circ \) or \( \frac{\pi}{2} \).

### Conclusion
From the calculations, we can conclude that the diagonals of the parallelogram intersect at an angle of \( \frac{\pi}{2} \).

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