To determine the slope of a line that is perpendicular to line [tex]\(m\)[/tex], we need to understand the relationship between the slopes of perpendicular lines.
Given:
- Line [tex]\(m\)[/tex] has a slope of [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p > 0\)[/tex], [tex]\(q > 0\)[/tex], and [tex]\(p \neq q\)[/tex].
### Step-by-Step Solution
1. Identify the slope of line [tex]\(m\)[/tex]:
The slope of line [tex]\(m\)[/tex] is given as [tex]\(\frac{p}{q}\)[/tex].
2. Concept of Perpendicular Slopes:
If two lines are perpendicular, then the product of their slopes is [tex]\(-1\)[/tex]. Mathematically, if [tex]\(m_1\)[/tex] is the slope of one line, and [tex]\(m_2\)[/tex] is the slope of a line perpendicular to it, then:
[tex]\[
m_1 \cdot m_2 = -1
\][/tex]
3. Determine the Perpendicular Slope:
- Let [tex]\(m_1 = \frac{p}{q}\)[/tex].
- We need to find [tex]\(m_2\)[/tex], the slope of the line perpendicular to line [tex]\(m\)[/tex].
Using the relationship for perpendicular slopes:
[tex]\[
\frac{p}{q} \cdot m_2 = -1
\][/tex]
4. Solve for [tex]\(m_2\)[/tex]:
[tex]\[
m_2 = -\frac{q}{p}
\][/tex]
### Conclusion:
The slope of a line that is perpendicular to line [tex]\(m\)[/tex], which has a slope of [tex]\(\frac{p}{q}\)[/tex], is:
[tex]\[
m_2 = -\frac{q}{p}
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{-\frac{q}{p}} \][/tex]
