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

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:

& \Rightarrow m=3 \\\ & \Rightarrow c=4 \\\ \end{aligned}$$ We have to calculate the slope of the perpendicular. Let the slope of the perpendicular be denoted by M. To find the slope we should apply the formula$$mM=-1$$. After we put the value of$$m=3$$in $$mM=-1$$, we get: $$\begin{aligned} & \Rightarrow mM=-1 \\\ & \Rightarrow 3M=-1 \\\ & \Rightarrow M=-\dfrac{1}{3} \\\ \end{aligned}$$ Now we have got the value of the slope of the perpendicular. So write the equation of the perpendicular using slope intercept form. $$\begin{aligned} & \Rightarrow y=Mx+C \\\ & \Rightarrow y=-\dfrac{1}{3}x+C.....\left( 2 \right) \\\ \end{aligned}$$ Now we have equation (2) where C is unknown. To find the value of C put $$x=3,y=5$$ in the equation (2) because it passes through the point $$\left( 3,5 \right)$$. $$\begin{aligned} & \Rightarrow y=-\dfrac{1}{3}x+C \\\ & \Rightarrow 5=-\dfrac{1}{3}\cdot 3+C \\\ & \Rightarrow 5=-1+C \\\ & \Rightarrow C=6 \\\ \end{aligned}$$ Put $$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.