To solve this problem, we need to find the slope of a line that is perpendicular to a given line [tex]\( m \)[/tex].
1. Identify the given slope of line [tex]\( m \)[/tex]:
We know that the slope of line [tex]\( m \)[/tex] is [tex]\(\frac{p}{q}\)[/tex], where [tex]\( p > 0 \)[/tex] and [tex]\( q > 0 \)[/tex].
2. Understand the concept of perpendicular slopes:
For two lines to be perpendicular, the slope of one line must be the negative reciprocal of the slope of the other line. If one line has a slope [tex]\( m_1 \)[/tex], the slope of a line perpendicular to it, [tex]\( m_2 \)[/tex], can be expressed as:
[tex]\[
m_2 = -\frac{1}{m_1}
\][/tex]
3. Apply this concept to our given slope:
- Our given slope [tex]\( m_1 \)[/tex] is [tex]\(\frac{p}{q}\)[/tex].
- The negative reciprocal of [tex]\(\frac{p}{q}\)[/tex] is:
[tex]\[
-\frac{1}{\left( \frac{p}{q} \right)} = -\frac{q}{p}
\][/tex]
4. Conclusion:
Therefore, the slope of a line that is perpendicular to the line with slope [tex]\(\frac{p}{q}\)[/tex] is:
[tex]\[
-\frac{q}{p}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{-\frac{q}{p}}
\][/tex]
