The area (in square units ) of the quadrilateral formed by two pairs of lines `l^(2) x^(2) - m^(2) y^(2) - n (lx + my) = 0` and `l^(2) x^(2)- m^(2) y^(2) + n (lx - my ) = 0` , is

Options

A. `(n^(2))/(2|lm|)`
B. `(n^(2))/(|m|)`
C. `(n)/(2|lm|)`
D. `(n^(2))/(4|lm|)`

Explanation

To find the area of the quadrilateral formed by the two pairs of lines given by the equations:

1. \( l^2 x^2 - m^2 y^2 - n(lx + my) = 0 \)
2. \( l^2 x^2 - m^2 y^2 + n(lx - my) = 0 \)

we can follow these steps:

### Step 1: Rewrite the equations
We start by rewriting the first equation:

\[ l^2 x^2 - m^2 y^2 - n(lx + my) = 0 \]

This can be factored into two linear equations:

\[ (lx + my)(lx - my) - n = 0 \]

Thus, the first pair of lines is:

- \( lx + my = 0 \) (Line 1)
- \( lx - my = n \) (Line 2)

Now, let's rewrite the second equation:

\[ l^2 x^2 - m^2 y^2 + n(lx - my) = 0 \]

This can also be factored into two linear equations:

\[ (lx - my)(lx + my) + n = 0 \]

Thus, the second pair of lines is:

- \( lx - my = 0 \) (Line 3)
- \( lx + my = -n \) (Line 4)

### Step 2: Identify coefficients
We can identify the coefficients from the equations of the lines:

- From Line 1: \( a_1 = l, b_1 = m, c_1 = 0 \)
- From Line 2: \( a_2 = l, b_2 = -m, c_2 = n \)
- From Line 3: \( a_3 = l, b_3 = -m, c_3 = 0 \)
- From Line 4: \( a_4 = l, b_4 = m, c_4 = -n \)

### Step 3: Use the area formula
The area \( A \) of the quadrilateral formed by these pairs of lines can be calculated using the formula:

\[ A = \frac{|c_1 - d_1| \cdot |c_2 - d_2|}{|a_1 b_2 - a_2 b_1|} \]

Where:
- \( c_1 = 0 \), \( d_1 = n \)
- \( c_2 = n \), \( d_2 = 0 \)

Substituting these values into the formula gives:

\[ A = \frac{|0 - n| \cdot |n - 0|}{|l \cdot (-m) - l \cdot m|} = \frac{n^2}{|-lm - lm|} = \frac{n^2}{|-2lm|} \]

Since we are interested in the area, we take the absolute value:

\[ A = \frac{n^2}{2lm} \]

### Final Answer
Thus, the area of the quadrilateral formed by the two pairs of lines is:

\[ \boxed{\frac{n^2}{2lm}} \]