Find the equation of the line passing through the point (-4,-5) and perpendicular to the line joining the points (1,2) and (5,6)

Options

A. a. x+y+17=0
B. b. 3x+2y+11=0
C. c. x+y+8=0
D. d. x-y+20=0

Answer

To find the equation of the line passing through the point (-4, -5) and perpendicular to the line joining the points (1, 2) and (5, 6), we can follow these steps:

### Step 1: Find the slope of the line joining the points (1, 2) and (5, 6).

The formula for the slope (m) between two points (x1, y1) and (x2, y2) is given by:
\[ m = \frac{y2 - y1}{x2 - x1} \]

Substituting the coordinates:
- (x1, y1) = (1, 2)
- (x2, y2) = (5, 6)

Calculating the slope:
\[ m = \frac{6 - 2}{5 - 1} = \frac{4}{4} = 1 \]

### Step 2: Determine the slope of the perpendicular line.

If two lines are perpendicular, the product of their slopes (m1 and m2) is -1. Therefore, if the slope of the first line (m1) is 1, the slope of the perpendicular line (m2) can be found as follows:
\[ m1 \cdot m2 = -1 \]
\[ 1 \cdot m2 = -1 \]
Thus,
\[ m2 = -1 \]

### Step 3: Use the point-slope form to find the equation of the line.

The point-slope form of the equation of a line is given by:
\[ y - y1 = m(x - x1) \]

Here, we have:
- Point (-4, -5) which gives us (x1, y1) = (-4, -5)
- Slope (m) = -1

Substituting these values into the point-slope form:
\[ y - (-5) = -1(x - (-4)) \]
\[ y + 5 = -1(x + 4) \]

### Step 4: Simplify the equation.

Distributing the -1:
\[ y + 5 = -x - 4 \]

Now, isolate y:
\[ y = -x - 4 - 5 \]
\[ y = -x - 9 \]

### Step 5: Write the equation in standard form.

To write the equation in standard form (Ax + By + C = 0):
\[ x + y + 9 = 0 \]

Thus, the equation of the line is:
\[ x + y + 9 = 0 \]