What is the equation of the line that is perpendicular to [tex]y+1=-3(x-5)[/tex] and passes through the point \((4,-6)\)?

A. [tex]y+6=-3(x-4)[/tex]
B. [tex]y-6=3(x+4)[/tex]
C. [tex]y-6=-\frac{1}{3}(x+4)[/tex]
D. [tex]y+6=\frac{1}{3}(x-4)[/tex]



Answer :

To determine the equation of a line that is perpendicular to the given line and passes through a specific point, we follow these steps:

1. Identify the slope of the given line:
The given line is \( y + 1 = -3(x - 5) \). This can be written in slope-intercept form \( y = mx + b \).

Start by expanding the equation:
[tex]\[ y + 1 = -3(x - 5) \][/tex]
[tex]\[ y + 1 = -3x + 15 \][/tex]
[tex]\[ y = -3x + 14 \][/tex]

Hence, the slope of the given line is \( -3 \).

2. Find the slope of the perpendicular line:
The slopes of perpendicular lines are negative reciprocals of each other. Therefore, the slope \( m \) of the line perpendicular to the given line is:
[tex]\[ m = -\frac{1}{-3} = \frac{1}{3} \][/tex]

3. Use the point-slope form to find the equation of the new line:
The point-slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.

We have \( m = \frac{1}{3} \) and the point \( (4, -6) \). Substitute these values into the point-slope form:
[tex]\[ y - (-6) = \frac{1}{3}(x - 4) \][/tex]
Simplify to:
[tex]\[ y + 6 = \frac{1}{3}(x - 4) \][/tex]

4. Identify the correct choice:
Among the given options, the equation \( y + 6 = \frac{1}{3}(x - 4) \) corresponds to option D.

Therefore, 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]

Thus, the correct answer is [tex]\( D \)[/tex].