To find the equations of the lines passing through the point [tex]\((4, -3)\)[/tex] that are parallel and perpendicular to the line given by [tex]\(y = -\frac{1}{2} x + 5\)[/tex], we follow these steps:
1. Identify the slope of the given line:
The given line has the equation [tex]\(y = -\frac{1}{2} x + 5\)[/tex]. The slope of this line is [tex]\(-\frac{1}{2}\)[/tex].
2. Equation of the parallel line:
Lines that are parallel share the same slope. Therefore, the parallel line must also have a slope of [tex]\(-\frac{1}{2}\)[/tex].
Using the point-slope form of a line, [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the point [tex]\((4, -3)\)[/tex] and [tex]\(m\)[/tex] is the slope:
[tex]\[
y - (-3) = -\frac{1}{2}(x - 4)
\][/tex]
Simplify to get it into slope-intercept form [tex]\(y = mx + c\)[/tex]:
[tex]\[
y + 3 = -\frac{1}{2}x + 2
\][/tex]
[tex]\[
y = -\frac{1}{2}x - 1
\][/tex]
Therefore, the equation of the parallel line is [tex]\(y = -\frac{1}{2} x - 1\)[/tex].
3. Equation of the perpendicular line:
Perpendicular lines have slopes that are negative reciprocals. Thus, the slope of the line perpendicular to the given line is:
[tex]\[
m_{\text{perpendicular}} = -\frac{1}{\left(-\frac{1}{2}\right)} = 2
\][/tex]
Again, using the point-slope form with the point [tex]\((4, -3)\)[/tex] and the slope [tex]\(2\)[/tex]:
[tex]\[
y - (-3) = 2(x - 4)
\][/tex]
Simplify to get it into slope-intercept form [tex]\(y = mx + c\)[/tex]:
[tex]\[
y + 3 = 2x - 8
\][/tex]
[tex]\[
y = 2x - 11
\][/tex]
Therefore, the equation of the perpendicular line is [tex]\(y = 2 x - 11\)[/tex].
Considering the options provided:
a. [tex]\(y = -\frac{1}{2} x - 1\)[/tex] is parallel
b. [tex]\(y = -\frac{1}{2} x - 5\)[/tex] is parallel [tex]\(y = 2 x - 11\)[/tex] is perpendicular
c. [tex]\(y = -\frac{1}{2} x - 1\)[/tex] is parallel [tex]\(y = \frac{1}{2} x + 5\)[/tex] is perpendicular [tex]\(y = -2 x + 5\)[/tex] is perpendicular
From the derived equations:
- The equation of the parallel line is [tex]\(y = -\frac{1}{2} x - 1\)[/tex], which matches option a and part of option c.
- The equation of the perpendicular line is [tex]\(y = 2 x - 11\)[/tex], which matches option b.
Thus, the correct option that best answers the question is:
a. [tex]\(y = -\frac{1}{2} x - 1\)[/tex] is parallel
