To determine the slope of a line perpendicular to line [tex]\( m \)[/tex], we start with some fundamental concepts about slopes of perpendicular lines.
Given:
- Line [tex]\( m \)[/tex] has a slope of [tex]\(\frac{p}{q}\)[/tex].
To find:
- The slope of a line perpendicular to line [tex]\( m \)[/tex].
### Step-by-Step Solution:
1. Identify the slope of the given line:
The slope of line [tex]\( m \)[/tex] is [tex]\(\frac{p}{q}\)[/tex].
2. Understand the properties of perpendicular lines:
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. This is known as the negative reciprocal relationship. If [tex]\( m_1 \)[/tex] is the slope of the first line and [tex]\( m_2 \)[/tex] is the slope of the line perpendicular to it, then:
[tex]\[
m_1 \times m_2 = -1
\][/tex]
3. Calculate the slope of the perpendicular line:
Let's denote the slope of the line perpendicular to line [tex]\( m \)[/tex] as [tex]\( m_{\perp} \)[/tex].
According to the property of perpendicular slopes:
[tex]\[
\frac{p}{q} \times m_{\perp} = -1
\][/tex]
To find [tex]\( m_{\perp} \)[/tex], solve for [tex]\( m_{\perp} \)[/tex]:
[tex]\[
m_{\perp} = -\frac{1}{\frac{p}{q}}
\][/tex]
4. Simplify the expression:
Simplifying the fraction:
[tex]\[
m_{\perp} = -\frac{q}{p}
\][/tex]
### Conclusion:
The slope of the line perpendicular to line [tex]\( m \)[/tex], which has a slope of [tex]\(\frac{p}{q}\)[/tex], is:
[tex]\[
-\frac{q}{p}
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{-\frac{q}{p}}
\][/tex]
So, the correct answer is option B.
