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A line [tex]\( m \)[/tex] has a [tex]\( y \)[/tex]-intercept of [tex]\( c \)[/tex] and a slope of [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \ \textgreater \ 0 \)[/tex], [tex]\( q \ \textgreater \ 0 \)[/tex], and [tex]\( p \neq q \)[/tex].

What is the slope of a line that is perpendicular to line [tex]\( m \)[/tex]?

A. [tex]\( -\frac{q}{p} \)[/tex]

B. [tex]\( -\frac{p}{q} \)[/tex]

C. [tex]\( \frac{p}{q} \)[/tex]

D. [tex]\( \frac{q}{p} \)[/tex]



Answer :

GinnyAnswer
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]

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