Question

How do you write the equation that represents the line perpendicular to $y = - 3x + 4$ and passing through the point $( - 1,1)$?

Answer 1

Hint: Given an equation of line and we have to find the other equation of a line which is perpendicular to the given line. As we have already given the line in the slope intercept form so it will be easy for us to find the slope of that line by comparing the equation by the form \[y = mx + c\]. Here $m$ is equal to slope and $c$ is Y intercept. On comparing we will get the slope of the line by using the relation that if two lines are perpendicular to each other then their slopes have relation ${m_1}{m_2} = - 1$. Here ${m_1}$ is the slope of the first line and ${m_2}$ is the slope of the second line. By using this relation we will find the slope of the second line. After finding the slope as we are given the point through which it passes. By using the point slope form we will find the equation.

Formula used:
$Y - {y_1} = m\left( {X - {x_1}} \right)$
Here, $m$= slope;
${y_1}$= y coordinate; ${x_1}$=x coordinate.

Complete step-by-step answer:
Step1: We are given an equation of line in the slope intercept form i.e. $y = - 3x + 4$. We will now compare it with the formula \[y = mx + c\]. On comparing we will find the slope of two which is perpendicular to this line. By using the relation of slope ${m_1}{m_2} = - 1$ here, ${m_1} = - 3$ and we have to find ${m_2}$ by substituting the value into it.
$ \Rightarrow - 3{m_2} = - 1$
$ \Rightarrow {m_2} = \dfrac{1}{3}$
Step2: Now we will find the equation of line which is perpendicular to the given line $y = - 3x + 4$ and it passes through the point $\left( { - 1,1} \right)$. By using the point slope formula we will get
$Y - {y_1} = m\left( {X - {x_1}} \right)$
Here ${y_1} = 1;x = - 1$ and $m = \dfrac{1}{3}$. Substituting this in the formula we will get:
$ \Rightarrow y - 1 = m\left( {x + 1} \right)$
$ \Rightarrow y - 1 = \dfrac{1}{3}\left( {x + 1} \right)$
$ \Rightarrow 3y - x = - 4$
On rearranging we will get:
$x - 3y = 4$ is the required equation of the line.

Hence the answer is $x - 3y = 4$

Note:
In such types of questions students mainly get confused in applying the formula. As they don't know which formula they have to apply. So when the equation of a line is given and we have to find another equation of a line then apply the concept of relationship between the slopes of two lines. Always remember that the slope of two parallel lines are equal and perpendicular lines the product of their slope is -1 and keep in mind that we can find the slope of any line from its slope intercept form equation. If a point is given through which line passes then use slope point formula.
Commit to memory:$Y - {y_1} = m\left( {X - {x_1}} \right)$