Find the equation of the line through the point (4, -5) and (a) parallel to (b) perpendicular to the line `3x+4y+5=0`.

Answer

To find the equations of the line through the point (4, -5) that is (a) parallel to and (b) perpendicular to the line given by the equation \(3x + 4y + 5 = 0\), we will follow these steps:

### Part (a): Equation of the line parallel to \(3x + 4y + 5 = 0\)

1. **Find the slope of the given line**:
The equation of the line is \(3x + 4y + 5 = 0\). We can rewrite it in slope-intercept form \(y = mx + b\) to find the slope \(m\).
\[ 4y = -3x - 5 \implies y = -\frac{3}{4}x - \frac{5}{4} \]
Thus, the slope \(m_1\) of the given line is \(-\frac{3}{4}\).

2. **Use the same slope for the parallel line**:
Since parallel lines have the same slope, the slope \(m\) of the line we are looking for is also \(-\frac{3}{4}\).

3. **Use the point-slope form to find the equation**:
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Here, \((x_1, y_1) = (4, -5)\) and \(m = -\frac{3}{4}\).
\[ y - (-5) = -\frac{3}{4}(x - 4) \]
Simplifying this:
\[ y + 5 = -\frac{3}{4}x + 3 \]
\[ y = -\frac{3}{4}x + 3 - 5 \]
\[ y = -\frac{3}{4}x - 2 \]

4. **Convert to standard form**:
To convert \(y = -\frac{3}{4}x - 2\) to standard form \(Ax + By + C = 0\):
\[ 3x + 4y + 8 = 0 \]

### Part (b): Equation of the line perpendicular to \(3x + 4y + 5 = 0\)

1. **Find the slope of the given line (already found)**:
The slope \(m_1\) of the line \(3x + 4y + 5 = 0\) is \(-\frac{3}{4}\).

2. **Find the slope of the perpendicular line**:
The product of the slopes of two perpendicular lines is \(-1\). Thus, if \(m_1 = -\frac{3}{4}\), then the slope \(m_2\) of the perpendicular line is:
\[ m_2 = \frac{4}{3} \]

3. **Use the point-slope form to find the equation**:
Again using the point-slope form:
\[ y - y_1 = m(x - x_1) \]
Here, \((x_1, y_1) = (4, -5)\) and \(m = \frac{4}{3}\):
\[ y - (-5) = \frac{4}{3}(x - 4) \]
Simplifying this:
\[ y + 5 = \frac{4}{3}x - \frac{16}{3} \]
\[ y = \frac{4}{3}x - \frac{16}{3} - 5 \]
\[ y = \frac{4}{3}x - \frac{16}{3} - \frac{15}{3} \]
\[ y = \frac{4}{3}x - \frac{31}{3} \]

4. **Convert to standard form**:
To convert \(y = \frac{4}{3}x - \frac{31}{3}\) to standard form:
\[ 4x - 3y - 31 = 0 \]

### Final Answers:
- (a) The equation of the line parallel to \(3x + 4y + 5 = 0\) is \(3x + 4y + 8 = 0\).
- (b) The equation of the line perpendicular to \(3x + 4y + 5 = 0\) is \(4x - 3y - 31 = 0\).