To determine the equation of the line that is perpendicular to the line \( y + 1 = -3(x - 5) \) and passes through the point \( (4, -6) \), let’s work through the problem step by step.
1. Identify the Slope of the Given Line:
The given line equation is in point-slope form: \( y + 1 = -3(x - 5) \).
Here, the slope (\( m_1 \)) of the given line is \(-3\).
2. Determine the Slope of the Perpendicular Line:
For two lines to be perpendicular, the product of their slopes must be \(-1\). If the slope of the given line is \(-3\), the slope of the perpendicular line (\( m_2 \)) can be found using the negative reciprocal:
[tex]\[
m_2 = \frac{1}{-3} = \frac{1}{3}
\][/tex]
3. Use Point-Slope Form to Write the Perpendicular Line Equation:
The point-slope form equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, \( (x_1, y_1) = (4, -6) \) and \( m = \frac{1}{3} \).
Substituting these into the form:
[tex]\[
y - (-6) = \frac{1}{3}(x - 4)
\][/tex]
Simplifying further:
[tex]\[
y + 6 = \frac{1}{3}(x - 4)
\][/tex]
4. Compare with Given Options:
- Option A: \( y + 6 = -3(x - 4) \)
- Option B: \( y - 6 = 3(x + 4) \)
- Option C: \( y - 6 = -\frac{1}{3}(x + 4) \)
- Option D: \( y + 6 = \frac{1}{3}(x - 4) \)
The correct form that matches our derived equation \( y + 6 = \frac{1}{3}(x - 4) \) is Option D.
Thus, the equation of the line that is perpendicular to \( y + 1 = -3(x - 5) \) and passes through the point \( (4, -6) \) is:
[tex]\[
\boxed{y + 6 = \frac{1}{3}(x - 4)}
\][/tex]
Hence, the correct answer is Option D.
