7.2 
Parallel Lines and Perpendicular Lines


If two straight lines are parallel, then their gradients are equal, that is \(m_1 = m_2\) , and vice versa.


Example:
Given that the following pair of straight lines are parallel, find the value of \(k\).


\(\begin{aligned} &3x+ky=2 \\\\ &\text{and} \\\\ &4y+x8=0 \end{aligned}\)

Write in the gradient form. 

\(\begin{aligned} 3x+ky&=2 \\\\ ky&=3x+2 \\\\\ y&=\dfrac{3}{k}x+\dfrac{2}{k} \\\\ \therefore m_1&=\dfrac{3}{k}.\end{aligned}\)
\(\begin{aligned} 4y+x8&=0 \\\\ 4y&=x+8 \\\\ y&=\dfrac{1}{4}x+2 \\\\ \therefore m_2&=\dfrac{1}{4}. \end{aligned}\)

Since the following pair of straight lines are parallel, hence 

\(m_1 = m_2.\) 

Therefore, 

\(\begin{aligned} \dfrac{3}{k}&=\dfrac{1}{4} \\\\ k&=12. \end{aligned}\)

If two straight lines are perpendicular, then the product of their gradients is \(1\), that is \(m_1m_2=1\), and vice versa. 

Example:
Given that the following pair of straight lines are perpendicular, find the value of \(p\).


\(\begin{aligned} &px+6y=8 \\\\ &\text{and} \\\\ &y6x=24 \end{aligned}\)
\(\begin{aligned}px+6y&=8\\\\ 6y&=px+8 \\\\ y&=\dfrac{p}{6}x+\dfrac{4}{3}\\\\ \therefore m_1&=\dfrac{p}{6}. \end{aligned}\)
\(\begin{aligned} y6x&=24 \\\\ y&=6x+24\\\\ \therefore m_2&=6. \end{aligned}\)

Since the following pair of straight lines are perpendicular, hence 

\(m_1m_2=1.\)
\(\begin{aligned} \left( \dfrac{p}{6} \right) \left(6 \right) &=1\\\\p&=1 \\\\ p&=1. \end{aligned}\)