The coordinates of the points A, B, C, D are (2, a), (3,5), (3,4) and (0, 6) respectively. If the lines AC and BD be perpendicular, then a = ?

Options

A. 7
B. 1
C. -1
D. -7

Answer

To find the value of \( a \) such that the lines AC and BD are perpendicular, we will follow these steps:

### Step 1: Identify the coordinates of points A, B, C, and D.
- Point A: \( (2, a) \)
- Point B: \( (3, 5) \)
- Point C: \( (3, 4) \)
- Point D: \( (0, 6) \)

### Step 2: Calculate the slope of line AC.
The slope \( m_1 \) of line AC can be calculated using the formula:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} \]
For points A and C:
- \( (x_1, y_1) = (2, a) \)
- \( (x_2, y_2) = (3, 4) \)

Substituting these values into the slope formula:
\[ m_1 = \frac{4 - a}{3 - 2} = 4 - a \]

### Step 3: Calculate the slope of line BD.
The slope \( m_2 \) of line BD can be calculated similarly:
For points B and D:
- \( (x_1, y_1) = (3, 5) \)
- \( (x_2, y_2) = (0, 6) \)

Using the slope formula:
\[ m_2 = \frac{6 - 5}{0 - 3} = \frac{1}{-3} = -\frac{1}{3} \]

### Step 4: Set up the equation for perpendicular lines.
For two lines to be perpendicular, the product of their slopes must equal -1:
\[ m_1 \cdot m_2 = -1 \]
Substituting the values of \( m_1 \) and \( m_2 \):
\[ (4 - a) \cdot \left(-\frac{1}{3}\right) = -1 \]

### Step 5: Solve for \( a \).
Multiplying both sides by -3 to eliminate the fraction:
\[ 4 - a = 3 \]
Now, solving for \( a \):
\[ -a = 3 - 4 \]
\[ -a = -1 \]
\[ a = 1 \]

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