The line passing through (-4, -2) and (2, -3) is perpendicular to the line passing through (a, 5) and (2, -1). Find a. 

Answer

To solve the problem, we need to find the value of \( a \) such that the line passing through the points \((-4, -2)\) and \((2, -3)\) is perpendicular to the line passing through the points \((a, 5)\) and \((2, -1)\).

### Step 1: Calculate the slope of the first line

The slope \( m_1 \) of the line passing through the points \((-4, -2)\) and \((2, -3)\) can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates:

\[ m_1 = \frac{-3 - (-2)}{2 - (-4)} = \frac{-3 + 2}{2 + 4} = \frac{-1}{6} \]

### Step 2: Calculate the slope of the second line

The slope \( m_2 \) of the line passing through the points \((a, 5)\) and \((2, -1)\) is given by:

\[ m_2 = \frac{-1 - 5}{2 - a} = \frac{-6}{2 - a} \]

### Step 3: Use the condition for perpendicular lines

Since the two lines are perpendicular, the product of their slopes must equal \(-1\):

\[ m_1 \cdot m_2 = -1 \]

Substituting the values of \( m_1 \) and \( m_2 \):

\[ \left(-\frac{1}{6}\right) \cdot \left(-\frac{6}{2 - a}\right) = -1 \]

### Step 4: Simplify the equation

This simplifies to:

\[ \frac{1}{6} \cdot \frac{6}{2 - a} = -1 \]

The \(6\) in the numerator and denominator cancels out:

\[ \frac{1}{2 - a} = -1 \]

### Step 5: Cross-multiply to solve for \( a \)

Cross-multiplying gives:

\[ 1 = -1(2 - a) \]

This simplifies to:

\[ 1 = -2 + a \]

### Step 6: Solve for \( a \)

Adding \(2\) to both sides:

\[ a = 1 + 2 = 3 \]

Thus, the value of \( a \) is \( 3 \).

### Final Answer

\[ \boxed{3} \]