Step 1: Understanding the Concept:
Two homogeneous equations in \( l, m, n \) typically represent two lines passing through the origin. To find the angle between them, we need to find the specific direction ratios \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \) or identify their dot product.
If the lines are perpendicular, the angle is \( \pi/2 \), and \( l_1l_2 + m_1m_2 + n_1n_2 = 0 \).
Key Formula or Approach:
Eliminate one variable from the equations to form a quadratic in the ratio of the remaining variables.
Step 2: Detailed Explanation:
From \( l + m + n = 0 \), we get \( n = -(l+m) \).
Substitute this into the second equation \( mn - 2ln + lm = 0 \):
\[ m(-(l+m)) - 2l(-(l+m)) + lm = 0 \]
\[ -lm - m^2 + 2l^2 + 2lm + lm = 0 \]
\[ 2l^2 + 2lm - m^2 = 0 \]
Divide by \( m^2 \) to create a quadratic in \( (l/m) \):
\[ 2\left(\frac{l}{m}\right)^2 + 2\left(\frac{l}{m}\right) - 1 = 0 \]
Let the two roots be \( r_1 = \frac{l_1}{m_1} \) and \( r_2 = \frac{l_2}{m_2} \).
From quadratic theory:
Product of roots \( r_1r_2 = \frac{c}{a} = \frac{-1}{2} \).
\[ \frac{l_1l_2}{m_1m_2} = -\frac{1}{2} \implies 2l_1l_2 = -m_1m_2 \implies 2l_1l_2 + m_1m_2 = 0 \dots (1) \]
Sum of roots \( r_1 + r_2 = \frac{-b}{a} = \frac{-2}{2} = -1 \).
\[ \frac{l_1}{m_1} + \frac{l_2}{m_2} = -1 \implies \frac{l_1m_2 + l_2m_1}{m_1m_2} = -1 \implies l_1m_2 + l_2m_1 = -m_1m_2 \dots (2) \]
Now consider \( n_1n_2 = (-(l_1+m_1))(-(l_2+m_2)) = l_1l_2 + l_1m_2 + l_2m_1 + m_1m_2 \).
Substitute from (2):
\[ n_1n_2 = l_1l_2 + (-m_1m_2) + m_1m_2 = l_1l_2 \]
The dot product for the angle is \( \sum l_1l_2 = l_1l_2 + m_1m_2 + n_1n_2 \).
\[ \sum l_1l_2 = l_1l_2 + m_1m_2 + l_1l_2 = 2l_1l_2 + m_1m_2 \]
From equation (1), this sum is zero.
Since the dot product of the direction ratios is zero, the lines are perpendicular.
Step 3: Final Answer:
The angle between the lines is \( 90^\circ \) or \( \pi/2 \).