To find the equation of the line that is perpendicular to a given line and passes through a specific point, we need to follow these steps:
1. Calculate the slope of the given line:
The given line passes through the points [tex]\((-4, -3)\)[/tex] and [tex]\( (4, 1) \)[/tex]. The slope [tex]\(m\)[/tex] of the line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]\( (x_1, y_1) = (-4, -3) \)[/tex] and [tex]\( (x_2, y_2) = (4, 1) \)[/tex].
[tex]\[
m = \frac{1 - (-3)}{4 - (-4)} = \frac{1 + 3}{4 + 4} = \frac{4}{8} = \frac{1}{2}
\][/tex]
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 original line’s slope. If the slope of the given line is [tex]\( \frac{1}{2} \)[/tex], then the slope of the perpendicular line is:
[tex]\[
m_{\perp} = -\frac{1}{\frac{1}{2}} = -2
\][/tex]
3. Write the equation of the perpendicular line in point-slope form:
The point-slope form of the equation of a line with slope [tex]\(m\)[/tex] passing through a point [tex]\((x_1, y_1)\)[/tex] is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
In this case, the perpendicular line passes through the point [tex]\((-4, 3)\)[/tex] with a slope of [tex]\( -2 \)[/tex]. Substituting these values into the point-slope form:
[tex]\[
y - 3 = -2(x + 4)
\][/tex]
So, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-4, 3)\)[/tex] in point-slope form is:
[tex]\[
y - 3 = -2(x + 4)
\][/tex]
Therefore, the correct equation from the given options is:
[tex]\[
y - 3 = -2(x + 4)
\][/tex]
Hence, the answer is [tex]\( \boxed{y - 3 = -2(x + 4)} \)[/tex].
