If the points (-2,6) and (4,8) is perpendicular to the line joining the points (8,12) and (x,24) then the value of x is

Options

A. (a) `2/7`
B. (b) `4`
C. (c) `3`
D. (d) `-4`

Answer

To solve the problem, we need to find the value of \( x \) such that the line joining the points (-2, 6) and (4, 8) is perpendicular to the line joining the points (8, 12) and (x, 24).

### Step-by-Step Solution:

1. **Identify the Points**:
Let the points be defined as follows:
- Point A: \( (-2, 6) \) (denote as \( (x_1, y_1) \))
- Point B: \( (4, 8) \) (denote as \( (x_2, y_2) \))
- Point C: \( (8, 12) \) (denote as \( (x_3, y_3) \))
- Point D: \( (x, 24) \) (denote as \( (x_4, y_4) \))

2. **Calculate the Slope of Line AB**:
The slope \( m_1 \) of line AB can be calculated using the formula:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 6}{4 - (-2)} = \frac{2}{6} = \frac{1}{3} \]

3. **Calculate the Slope of Line CD**:
The slope \( m_2 \) of line CD can be calculated using the formula:
\[ m_2 = \frac{y_4 - y_3}{x_4 - x_3} = \frac{24 - 12}{x - 8} = \frac{12}{x - 8} \]

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 we found:
\[ \frac{1}{3} \cdot \frac{12}{x - 8} = -1 \]

5. **Solve for x**:
Multiply both sides by \( 3(x - 8) \) to eliminate the fraction:
\[ 12 = -3(x - 8) \]
Expanding the right side:
\[ 12 = -3x + 24 \]
Rearranging gives:
\[ -3x = 12 - 24 \]
\[ -3x = -12 \]
Dividing by -3:
\[ x = 4 \]

### Conclusion:
The value of \( x \) is \( 4 \).