The combined equation of the lines `L_(1)` and `L_(2)` is `2x^(2)+6xy+y^(2)=0` and that lines `L_(3)` and `L_(4)` is `4x^(2)+18xy+y^(2)=0`. If the angle between `L_(1)` and `L_(4)` be `alpha`, then the angle between `L_(2)` and `L_(3)` will be

Options

A. `(pi)/(2)-alpha`
B. `2alpha`
C. `(pi)/(4)+alpha`
D. `alpha`

Explanation

To solve the problem, we need to find the angle between the lines \( L_2 \) and \( L_3 \) given the combined equations of the lines \( L_1 \) and \( L_2 \), and \( L_3 \) and \( L_4 \).

### Step-by-step Solution:

1. **Identify the equations of the lines:**
- The combined equation of lines \( L_1 \) and \( L_2 \) is given as:
\[ 2x^2 + 6xy + y^2 = 0 \]
- The combined equation of lines \( L_3 \) and \( L_4 \) is given as:
\[ 4x^2 + 18xy + y^2 = 0 \]

2. **Determine the coefficients for the first pair of lines:**
- For the equation \( 2x^2 + 6xy + y^2 = 0 \), we can identify:
- \( a_1 = 2 \)
- \( b_1 = 1 \)
- \( h_1 = 3 \)

3. **Determine the coefficients for the second pair of lines:**
- For the equation \( 4x^2 + 18xy + y^2 = 0 \), we can identify:
- \( a_2 = 4 \)
- \( b_2 = 1 \)
- \( h_2 = 9 \)

4. **Find the angle bisector equations:**
- The equation of the angle bisector between lines \( L_1 \) and \( L_2 \) can be expressed as:
\[ \frac{x^2 - y^2}{a_1 - b_1} = \frac{xy}{h_1} \]
- Substituting the values:
\[ \frac{x^2 - y^2}{2 - 1} = \frac{xy}{3} \implies 3(x^2 - y^2) = xy \]
- Rearranging gives:
\[ 3x^2 - xy - 3y^2 = 0 \quad \text{(Equation 1)} \]

5. **Find the angle bisector equation for \( L_3 \) and \( L_4 \):**
- The equation of the angle bisector between lines \( L_3 \) and \( L_4 \) is:
\[ \frac{x^2 - y^2}{a_2 - b_2} = \frac{xy}{h_2} \]
- Substituting the values:
\[ \frac{x^2 - y^2}{4 - 1} = \frac{xy}{9} \implies 9(x^2 - y^2) = 3xy \]
- Rearranging gives:
\[ 3x^2 - xy - 3y^2 = 0 \quad \text{(Equation 2)} \]

6. **Comparing the two equations:**
- We see that Equation 1 and Equation 2 are identical:
\[ 3x^2 - xy - 3y^2 = 0 \]
- This indicates that the angle bisectors of both pairs of lines are the same, meaning the lines are equally inclined.

7. **Conclusion about the angles:**
- If the angle between \( L_1 \) and \( L_4 \) is \( \alpha \), then the angle between \( L_2 \) and \( L_3 \) must also be \( \alpha \).

### Final Answer:
The angle between \( L_2 \) and \( L_3 \) is \( \alpha \).

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