iphvne
Answered

What is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point \((-4,-3)\)?

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



Answer :

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

1. Identify the slope of the given line:
The equation of the given line is \( y + 3 = -4(x + 4) \). We can rewrite this in slope-intercept form ( \( y = mx + b \) ) to easily identify the slope \( m \).

Let's do that by isolating \( y \):
[tex]\[ y + 3 = -4(x + 4) \][/tex]
[tex]\[ y + 3 = -4x - 16 \][/tex]
[tex]\[ y = -4x - 16 - 3 \][/tex]
[tex]\[ y = -4x - 19 \][/tex]
Thus, the slope \( m \) of the given line is \( -4 \).

2. Determine the slope of the perpendicular line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. The negative reciprocal of \( -4 \) is:
[tex]\[ \frac{1}{-(-4)} = \frac{1}{4} \][/tex]
So, the slope of the perpendicular line is \( \frac{1}{4} \).

3. Use the point-slope form to write the equation of the perpendicular line:
The point-slope form of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where \( m \) is the slope of the line and \( (x_1, y_1) \) is a point on the line.

We are given the point \( (-4, -3) \) and the slope \( \frac{1}{4} \). Substituting these values into the point-slope form, we get:
[tex]\[ y - (-3) = \frac{1}{4}(x - (-4)) \][/tex]
Simplifying the expression:
[tex]\[ y + 3 = \frac{1}{4}(x + 4) \][/tex]

Therefore, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-4, -3)\)[/tex] is [tex]\( y + 3 = \frac{1}{4}(x + 4) \)[/tex].