If the mid-point of the line joining (3, 4) and (p, 7) is (x, y) and 2x + 2y + 1 = 0, then what will be the value of p?

Options

A. 15
B. `(-17)/(2)`
C. -15
D. `(17)/(2)`

Answer

To solve the problem step by step, we can follow these instructions:

1. **Identify the Points**: We have two points: \( A(3, 4) \) and \( B(p, 7) \).

2. **Use the Midpoint Formula**: The midpoint \( M(x, y) \) of the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( x_1 = 3 \), \( y_1 = 4 \), \( x_2 = p \), and \( y_2 = 7 \).

3. **Calculate the Midpoint Coordinates**:
- The x-coordinate of the midpoint:
\[ x = \frac{3 + p}{2} \]
- The y-coordinate of the midpoint:
\[ y = \frac{4 + 7}{2} = \frac{11}{2} \]

4. **Substitute Midpoint Coordinates into the Given Equation**: We know from the problem that \( 2x + 2y + 1 = 0 \). Substitute \( x \) and \( y \) into this equation:
\[ 2\left(\frac{3 + p}{2}\right) + 2\left(\frac{11}{2}\right) + 1 = 0 \]

5. **Simplify the Equation**:
- The first term simplifies to \( 3 + p \).
- The second term simplifies to \( 11 \).
- So we have:
\[ 3 + p + 11 + 1 = 0 \]
- Combine like terms:
\[ p + 15 = 0 \]

6. **Solve for \( p \)**:
\[ p = -15 \]

Thus, the value of \( p \) is \( -15 \).