To determine which equation represents the line that is perpendicular to [tex]\( y = \frac{1}{6} \)[/tex] and passes through the point [tex]\((-8, -2)\)[/tex], let's follow the steps below:
1. Analyze the given equation [tex]\( y = \frac{1}{6} \)[/tex].
- This is a horizontal line where [tex]\( y \)[/tex] is always [tex]\( \frac{1}{6} \)[/tex] regardless of the value of [tex]\( x \)[/tex].
2. Determine the slope of the given line.
- Since the line [tex]\( y = \frac{1}{6} \)[/tex] is horizontal, its slope is [tex]\( 0 \)[/tex].
3. Identify the slope of the perpendicular line.
- A line that is perpendicular to a horizontal line must be vertical because the slope of a vertical line is undefined.
4. Find the equation of the vertical line.
- A vertical line has the equation [tex]\( x = c \)[/tex], where [tex]\( c \)[/tex] is the [tex]\( x \)[/tex]-coordinate of any point on the line.
5. Use the given point [tex]\((-8, -2)\)[/tex] to find the equation of the perpendicular line.
- The [tex]\( x \)[/tex]-coordinate of the point [tex]\((-8, -2)\)[/tex] is [tex]\(-8\)[/tex].
Therefore, the equation of the line that is perpendicular to [tex]\( y = \frac{1}{6} \)[/tex] and passes through the point [tex]\((-8, -2)\)[/tex] is:
[tex]\[ x = -8 \][/tex]
Thus, the correct answer is:
[tex]\[ x = -8 \][/tex]
