The sum of all the values of p for which the lines `x+y-1=0, px+4y+2=0` and `4x+py+7=0` are concurrent is euqal to

Options

A. 0
B. `-9`
C. `-13`
D. 3

Answer

To determine the sum of all values of \( p \) for which the lines \( x + y - 1 = 0 \), \( px + 4y + 2 = 0 \), and \( 4x + py + 7 = 0 \) are concurrent, we can use the condition for concurrency of three lines.

### Step-by-Step Solution:

1. **Identify the equations of the lines**:
- Line 1: \( x + y - 1 = 0 \) (which can be rewritten as \( 1x + 1y - 1 = 0 \))
- Line 2: \( px + 4y + 2 = 0 \) (which can be rewritten as \( px + 4y + 2 = 0 \))
- Line 3: \( 4x + py + 7 = 0 \) (which can be rewritten as \( 4x + py + 7 = 0 \))

2. **Set up the determinant for concurrency**:
The lines are concurrent if the following determinant is equal to zero:
\[ \begin{vmatrix} 1 & 1 & -1 \\ p & 4 & -2 \\ 4 & p & -7 \end{vmatrix} = 0 \]

3. **Calculate the determinant**:
Expanding the determinant:
\[ = 1 \left( 4 \cdot (-7) - (-2) \cdot p \right) - 1 \left( p \cdot (-7) - (-2) \cdot 4 \right) + (-1) \left( p \cdot p - 4 \cdot 4 \right) \]
Simplifying each term:
\[ = 1(-28 + 2p) - 1(-7p + 8) - (p^2 - 16) \]
\[ = -28 + 2p + 7p - 8 - p^2 + 16 \]
Combining like terms:
\[ = -p^2 + 9p - 20 \]

4. **Set the determinant to zero**:
\[ -p^2 + 9p - 20 = 0 \]
Multiplying through by -1:
\[ p^2 - 9p + 20 = 0 \]

5. **Factor the quadratic equation**:
\[ (p - 4)(p - 5) = 0 \]
Thus, the solutions for \( p \) are:
\[ p = 4 \quad \text{or} \quad p = 5 \]

6. **Calculate the sum of the values of \( p \)**:
\[ \text{Sum} = 4 + 5 = 9 \]

### Final Answer:
The sum of all values of \( p \) for which the lines are concurrent is \( 9 \).