Question

How do I write the equation of the line that is perpendicular to \[y=3x+4\] and goes through the point\[\left( 3,5 \right)\]?

Answer 1

Hint: We are given with an equation of a line. First of all convert it into the slope intercept form. To write the equation of its perpendicular we should know that the product of the line and its perpendicular is \[-1\]. We should find the slope of the given line then to find the slope of the perpendicular divide the slope of the equation by \[-1\]. Now after getting the slope, put the values of the given coordinates and calculate the value of the constant.

Complete step by step solution:
We have our equation with us that is \[y=3x+4.....\left( 1 \right)\].
The equation is already in the slope intercept form. So let us compare it with\[y=mx+c\]and write the values of slope $m$ and the constant $c$. After comparing we get:
\[\begin{align}
  & \Rightarrow m=3 \\
 & \Rightarrow c=4 \\
\end{align}\]
We have to calculate the slope of the perpendicular. Let the slope of the perpendicular be denoted by M. To find the slope we should apply the formula\[mM=-1\]. After we put the value of\[m=3\]in \[mM=-1\], we get:
\[\begin{align}
  & \Rightarrow mM=-1 \\
 & \Rightarrow 3M=-1 \\
 & \Rightarrow M=-\dfrac{1}{3} \\
\end{align}\]
Now we have got the value of the slope of the perpendicular. So write the equation of the perpendicular using slope intercept form.
\[\begin{align}
  & \Rightarrow y=Mx+C \\
 & \Rightarrow y=-\dfrac{1}{3}x+C.....\left( 2 \right) \\
\end{align}\]
Now we have equation (2) where C is unknown. To find the value of C put \[x=3,y=5\] in the equation (2) because it passes through the point \[\left( 3,5 \right)\].
\[\begin{align}
  & \Rightarrow y=-\dfrac{1}{3}x+C \\
 & \Rightarrow 5=-\dfrac{1}{3}\cdot 3+C \\
 & \Rightarrow 5=-1+C \\
 & \Rightarrow C=6 \\
\end{align}\]
Put \[C=6\]in \[y=-\dfrac{1}{3}x+C\]to get\[y=-\dfrac{1}{3}x+6\]. Hence the equation of the perpendicular is\[y=-\dfrac{1}{3}x+6\].

Note:
While solving such type of questions we should always remember that slope of the line perpendicular to the given line is always \[-\dfrac{1}{\text{slope of the perpendicular line}}\]of the slope of the equation but in case of parallel lines, the slope of the parallel lines are always equal.