If the lines lx + my + n = 0, mx + ny + l = 0 and nx + ly + m = 0 are concurrent then

Options

A. l + m + n = 0
B. l- m - n = 0
C. l + m - n = 0
D. m+ n - l = 0

Answer

To determine the condition under which the lines \( lx + my + n = 0 \), \( mx + ny + l = 0 \), and \( nx + ly + m = 0 \) are concurrent, we can use the concept of the determinant of the coefficients of the lines.

### Step-by-Step Solution:

1. **Write the equations in standard form**:
The equations of the lines are given as:
\[ l x + m y + n = 0 \]
\[ m x + n y + l = 0 \]
\[ n x + l y + m = 0 \]

2. **Set up the determinant**:
For the lines to be concurrent, the determinant formed by the coefficients of \(x\), \(y\), and the constant terms must be zero. The determinant can be expressed as:
\[ \begin{vmatrix} l & m & n \\ m & n & l \\ n & l & m \end{vmatrix} \]

3. **Calculate the determinant**:
We can calculate the determinant using the formula for a 3x3 matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our matrix:
\[ D = l(nm - l^2) - m(m^2 - nl) + n(ml - n^2) \]

4. **Simplify the determinant**:
Expanding the determinant:
\[ D = l(nm - l^2) - m(m^2 - nl) + n(ml - n^2) \]
After simplification, we will find that:
\[ D = lnm + nml + ml^2 - l^3 - m^3 - n^3 \]

5. **Set the determinant to zero**:
For the lines to be concurrent, we set the determinant equal to zero:
\[ lnm + nml + ml^2 - l^3 - m^3 - n^3 = 0 \]

6. **Derive the condition**:
After further simplification, we find that one of the conditions that must hold is:
\[ l + m + n = 0 \]

### Conclusion:
Thus, the lines are concurrent if:
\[ l + m + n = 0 \]