Step 1: Concept:
Determine the line equation given a point and a perpendicular line. The core concept involves the slope relationship between perpendicular lines.
Step 2: Method:
1. Find the slope (\(m_1\)) of the given line, using \(m = -A/B\) for \(Ax + By + C = 0\).
2. Calculate the perpendicular line's slope (\(m_2\)) using \(m_2 = -1/m_1\).
3. Apply the point-slope form: \(y - y_1 = m_2(x - x_1)\).
4. Convert to the general form: \(Ax + By + C = 0\).
Step 3: Detailed Solution:
1. Find the given line's slope.
Given line: \(3x + 4y + 5 = 0\).
Slope \(m_1 = -\frac{A}{B} = -\frac{3}{4}\).
2. Find the perpendicular line's slope.
Required line's slope, \(m_2\), is the negative reciprocal of \(m_1\).
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{(-3/4)} = \frac{4}{3} \]
3. Use the point-slope form.
The line passes through \((x_1, y_1) = (4, -5)\) with \(m_2 = 4/3\).
\[ y - y_1 = m_2(x - x_1) \]
\[ y - (-5) = \frac{4}{3}(x - 4) \]
\[ y + 5 = \frac{4}{3}(x - 4) \]
4. Convert to general form.
Multiply by 3:
\[ 3(y + 5) = 4(x - 4) \]
\[ 3y + 15 = 4x - 16 \]
Rearrange:
\[ 0 = 4x - 3y - 16 - 15 \]
\[ 4x - 3y - 31 = 0 \]
Step 4: Answer:
The line equation is \(4x - 3y - 31 = 0\), which is option (1).