If the two liens px + 6y + 3 = 0 and 2x + qy+ 3 = 0 have infinite solutions then find the value of p and q respectively

Options

A. 2, 6
B. 6, 2
C. 2, 2
D. 6, 6

Answer

To solve the problem, we need to determine the values of \( p \) and \( q \) such that the lines given by the equations \( px + 6y + 3 = 0 \) and \( 2x + qy + 3 = 0 \) have infinite solutions. This occurs when the two lines are coincident, which means that the ratios of the coefficients of \( x \), \( y \), and the constant terms must be equal.

### Step-by-step Solution:

1. **Identify the Coefficients**:
- For the first line \( px + 6y + 3 = 0 \):
- Coefficient of \( x \) (A1) = \( p \)
- Coefficient of \( y \) (B1) = \( 6 \)
- Constant term (C1) = \( 3 \)
- For the second line \( 2x + qy + 3 = 0 \):
- Coefficient of \( x \) (A2) = \( 2 \)
- Coefficient of \( y \) (B2) = \( q \)
- Constant term (C2) = \( 3 \)

2. **Set Up the Ratios**:
- For the lines to have infinite solutions, the following ratios must be equal:
\[ \frac{A1}{A2} = \frac{B1}{B2} = \frac{C1}{C2} \]
- This gives us two equations to work with:
\[ \frac{p}{2} = \frac{3}{3} \quad \text{(1)} \]
\[ \frac{6}{q} = \frac{3}{3} \quad \text{(2)} \]

3. **Solve for \( p \)**:
- From equation (1):
\[ \frac{p}{2} = 1 \implies p = 2 \]

4. **Solve for \( q \)**:
- From equation (2):
\[ \frac{6}{q} = 1 \implies q = 6 \]

5. **Final Values**:
- The values of \( p \) and \( q \) are:
\[ p = 2, \quad q = 6 \]

### Conclusion:
Thus, the values of \( p \) and \( q \) that ensure the two lines have infinite solutions are \( p = 2 \) and \( q = 6 \).

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