The direction cosines of two lines are connected by relation `l+m+n=0` and 4l is the harmonic mean between m and n.
Then,

Options

A. `(l_(1)/l_(2))+m_(1)/m_(2)+n_(1)/n_(2)=-3//2`
B. `l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)=-1/2`
C. `l_(1)m_(1)n_(1)+l_(2)m_(2)n_(2)=-sqrt(6)//9`
D. `(l_(1)+l_(2))(m_(1)+m_(2))(n_(1)+n_(2))=sqrt(6)/18`

Explanation

To solve the problem, we need to find the direction cosines of two lines given the conditions \( l + m + n = 0 \) and \( 4l \) is the harmonic mean of \( m \) and \( n \).

### Step-by-Step Solution:

1. **Understanding the Harmonic Mean**:
The harmonic mean \( H \) of two numbers \( m \) and \( n \) is given by:
\[ H = \frac{2mn}{m+n} \]
According to the problem, we have:
\[ 4l = H \implies 4l = \frac{2mn}{m+n} \]

2. **Cross Multiplying**:
We can cross-multiply to eliminate the fraction:
\[ 4l(m + n) = 2mn \]
Simplifying gives:
\[ 4lm + 4ln = 2mn \]

3. **Substituting \( n \)**:
From the relation \( l + m + n = 0 \), we can express \( n \) in terms of \( l \) and \( m \):
\[ n = -l - m \]
Substitute this into the equation:
\[ 4lm + 4l(-l - m) = 2m(-l - m) \]
Simplifying this:
\[ 4lm - 4l^2 - 4lm = -2ml - 2m^2 \]
This reduces to:
\[ -4l^2 = -2ml - 2m^2 \]
or:
\[ 4l^2 = 2ml + 2m^2 \]

4. **Dividing by 2**:
Dividing the entire equation by 2:
\[ 2l^2 = ml + m^2 \]
Rearranging gives:
\[ 2l^2 - ml - m^2 = 0 \]

5. **Using the Quadratic Formula**:
This is a quadratic equation in \( l \). We can use the quadratic formula \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2 \), \( b = -m \), and \( c = -m^2 \):
\[ l = \frac{m \pm \sqrt{(-m)^2 - 4 \cdot 2 \cdot (-m^2)}}{2 \cdot 2} \]
Simplifying:
\[ l = \frac{m \pm \sqrt{m^2 + 8m^2}}{4} = \frac{m \pm 3m}{4} \]

6. **Finding Values of \( l \)**:
This gives us two possible values for \( l \):
\[ l = \frac{4m}{4} = m \quad \text{(1)} \]
\[ l = \frac{-2m}{4} = -\frac{m}{2} \quad \text{(2)} \]

7. **Finding Corresponding \( m \) and \( n \)**:
For case (1):
\[ l = m \implies n = -l - m = -2m \]
For case (2):
\[ l = -\frac{m}{2} \implies n = -l - m = \frac{m}{2} - m = -\frac{m}{2} \]

8. **Finding Direction Cosines**:
Now we can find the direction cosines for both cases:
- For case (1): \( (l, m, n) = (m, m, -2m) \)
- For case (2): \( (l, m, n) = \left(-\frac{m}{2}, m, -\frac{m}{2}\right) \)

9. **Normalizing Direction Cosines**:
Normalize these direction cosines to ensure they satisfy \( l^2 + m^2 + n^2 = 1 \).