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.
Let's start with the given information:
- Line [tex]\( m \)[/tex] has a slope of [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p > 0 \)[/tex] and [tex]\( q > 0 \)[/tex].
When two lines are perpendicular, the product of their slopes is [tex]\(-1\)[/tex]. In other words, if the slope of one line is [tex]\( m_1 \)[/tex] and the slope of the line perpendicular to it is [tex]\( m_2 \)[/tex], then:
[tex]\[ m_1 \times m_2 = -1 \][/tex]
Here, the slope of line [tex]\( m \)[/tex] is [tex]\( \frac{p}{q} \)[/tex]. Let's denote this as [tex]\( m_1 \)[/tex]:
[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 between perpendicular slopes:
[tex]\[ \left(\frac{p}{q}\right) \times m_2 = -1 \][/tex]
To solve for [tex]\( m_2 \)[/tex], divide both sides of the equation by [tex]\( \frac{p}{q} \)[/tex]:
[tex]\[ m_2 = -\frac{1}{\left(\frac{p}{q}\right)} \][/tex]
Since dividing by a fraction is the same as multiplying by its reciprocal, this simplifies to:
[tex]\[ m_2 = -\frac{q}{p} \][/tex]
Therefore, the slope of the line perpendicular to line [tex]\( m \)[/tex] is [tex]\( -\frac{q}{p} \)[/tex].
Among the given choices:
A. [tex]\( \frac{q}{p} \)[/tex]
B. [tex]\( -\frac{q}{p} \)[/tex]
C. [tex]\( \frac{p}{q} \)[/tex]
D. [tex]\( -\frac{p}{q} \)[/tex]
The correct answer is:
B. [tex]\( -\frac{q}{p} \)[/tex]
