To determine the slope of a line [tex]\( n \)[/tex] that is perpendicular to another line [tex]\( k \)[/tex] with a given slope, we use an important geometric principle about perpendicular lines.
When two lines are perpendicular, the slope of one line is the negative reciprocal of the slope of the other line.
Given:
- The slope ([tex]\( m_k \)[/tex]) of line [tex]\( k \)[/tex] is [tex]\( -6 \)[/tex].
We need to find the slope ([tex]\( m_n \)[/tex]) of line [tex]\( n \)[/tex] which is perpendicular to line [tex]\( k \)[/tex].
Step-by-Step Solution:
1. Understand the Concept of Negative Reciprocals:
- If two lines are perpendicular, the product of their slopes is [tex]\( -1 \)[/tex]. This means if [tex]\( m_k \)[/tex] is the slope of line [tex]\( k \)[/tex], then [tex]\( m_n \)[/tex] (the slope of the perpendicular line) is [tex]\( -\frac{1}{m_k} \)[/tex].
2. Apply the Negative Reciprocal:
- Here, the slope of line [tex]\( k \)[/tex] is [tex]\( -6 \)[/tex].
- To find the slope of line [tex]\( n \)[/tex], we take the negative reciprocal of [tex]\( -6 \)[/tex].
3. Calculate:
[tex]\[
m_n = -\frac{1}{-6} = \frac{1}{6}
\][/tex]
4. Simplification:
- The reciprocal of [tex]\( -6 \)[/tex] is [tex]\( -\frac{1}{6} \)[/tex], and taking the negative reciprocal transforms [tex]\( -\frac{1}{6} \)[/tex] into [tex]\(\frac{1}{6}\)[/tex].
Therefore, the slope of line [tex]\( n \)[/tex], which is perpendicular to line [tex]\( k \)[/tex] with a slope of [tex]\( -6 \)[/tex], is [tex]\( \frac{1}{6} \)[/tex].
So, the correct answer is:
[tex]\[
\boxed{\frac{1}{6}}
\][/tex]
