Find the slope of a line perpendicular to the line which passes through each pair of the following points (-k, h) and (b, -f)

Answer

To find the slope of a line perpendicular to the line that passes through the points (-k, h) and (b, -f), we can follow these steps:

### Step 1: Identify the Points
We have two points:
- Point 1: \( (x_1, y_1) = (-k, h) \)
- Point 2: \( (x_2, y_2) = (b, -f) \)

### Step 2: Use the Slope Formula
The formula for the slope \( m \) of a line passing through two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

### Step 3: Substitute the Values
Substituting the values of the points into the slope formula:
\[ m = \frac{-f - h}{b - (-k)} \]
This simplifies to:
\[ m = \frac{-f - h}{b + k} \]

### Step 4: Simplify the Slope
We can express the slope as:
\[ m = \frac{-(f + h)}{b + k} \]
Let’s denote this slope as \( m_1 \):
\[ m_1 = \frac{-(f + h)}{b + k} \]

### Step 5: Find the Slope of the Perpendicular Line
For two lines to be perpendicular, the product of their slopes \( m_1 \) and \( m_2 \) must equal -1:
\[ m_1 \cdot m_2 = -1 \]
Substituting \( m_1 \):
\[ \frac{-(f + h)}{b + k} \cdot m_2 = -1 \]

### Step 6: Solve for \( m_2 \)
To find \( m_2 \), we rearrange the equation:
\[ m_2 = \frac{(b + k)}{(f + h)} \]

### Final Result
Thus, the slope of the line that is perpendicular to the line passing through the points (-k, h) and (b, -f) is:
\[ m_2 = \frac{b + k}{f + h} \]