If one of the lines given by the equation `2x^(2)+pxy+3y^(2)=0` coincide with one of those given by `2x^(2)+qxy-3y^(2)=0` and the other lines represented by them are perpendicular , then value of `p+q`is

Options

A. 6
B. -6
C. -7
D. 7

Explanation

To solve the problem, we need to analyze the given equations of the lines and use the conditions provided.

### Step-by-Step Solution:

1. **Identify the Given Equations:**
The equations of the lines are:
- \( 2x^2 + pxy + 3y^2 = 0 \) (Equation 1)
- \( 2x^2 + qxy - 3y^2 = 0 \) (Equation 2)

2. **Condition of Coinciding Lines:**
For one line from Equation 1 to coincide with one line from Equation 2, the coefficients of the corresponding terms must be proportional. This gives us the following relationship:
\[ \frac{p}{q} = \frac{3}{-3} \implies p = -q \]

3. **Condition of Perpendicular Lines:**
The other lines represented by these equations must be perpendicular. The slopes \( m \) and \( m' \) of the lines can be derived from the equations. The condition for perpendicularity is given by:
\[ m \cdot m' = -1 \]
From the general form of the lines, we can express the slopes in terms of \( p \) and \( q \):
- From Equation 1: \( m + m' = -\frac{p}{3} \) and \( mm' = \frac{2}{3} \)
- From Equation 2: \( m + m' = \frac{q}{3} \) and \( mm' = -\frac{2}{3} \)

4. **Using the Coinciding Condition:**
Since we established that \( p = -q \), we can substitute \( q = -p \) into the perpendicularity condition:
\[ m + m' = -\frac{p}{3} \quad \text{and} \quad m + m' = \frac{-p}{3} \]
This means both equations are consistent.

5. **Finding the Values of p and q:**
From the product of slopes:
\[ mm' = \frac{2}{3} \quad \text{and} \quad mm' = -\frac{2}{3} \]
This leads to:
\[ -\frac{p^2}{9} = -\frac{2}{3} \implies p^2 = 6 \implies p = \sqrt{6} \text{ or } p = -\sqrt{6} \]
Since \( q = -p \), we have:
- If \( p = \sqrt{6} \), then \( q = -\sqrt{6} \)
- If \( p = -\sqrt{6} \), then \( q = \sqrt{6} \)

6. **Calculating \( p + q \):**
In both cases:
\[ p + q = \sqrt{6} - \sqrt{6} = 0 \]
or
\[ p + q = -\sqrt{6} + \sqrt{6} = 0 \]

### Final Answer:
The value of \( p + q \) is \( 0 \).