The straight line joining (1, 2) and (2, -2) is perpendicular to the line joining (8, 2) and (4, p). What will be the value of p?

Options

A. -1
B. 1
C. 3
D. None of these

Answer

To solve the problem, we need to find the value of \( p \) such that the line joining the points \( (1, 2) \) and \( (2, -2) \) is perpendicular to the line joining the points \( (8, 2) \) and \( (4, p) \).

### Step-by-Step Solution:

1. **Identify the Points:**
- Let \( P(1, 2) \) and \( Q(2, -2) \) be the first line.
- Let \( A(8, 2) \) and \( B(4, p) \) be the second line.

2. **Calculate the Slope of Line PQ:**
- The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- For points \( P(1, 2) \) and \( Q(2, -2) \):
\[ m_1 = \frac{-2 - 2}{2 - 1} = \frac{-4}{1} = -4 \]

3. **Calculate the Slope of Line AB:**
- For points \( A(8, 2) \) and \( B(4, p) \):
\[ m_2 = \frac{p - 2}{4 - 8} = \frac{p - 2}{-4} = \frac{2 - p}{4} \]

4. **Set Up the Perpendicular Condition:**
- Since the lines are perpendicular, the product of their slopes must equal \(-1\):
\[ m_1 \cdot m_2 = -1 \]
- Substituting the slopes:
\[ -4 \cdot \frac{2 - p}{4} = -1 \]

5. **Simplify the Equation:**
- Multiply both sides by 4 to eliminate the fraction:
\[ -4(2 - p) = -4 \]
- Simplifying gives:
\[ 2 - p = 1 \]

6. **Solve for \( p \):**
- Rearranging the equation:
\[ -p = 1 - 2 \]
\[ -p = -1 \]
\[ p = 1 \]

### Conclusion:
The value of \( p \) is \( 1 \).