Question

A line passes through $\left( {2,2} \right)$ and is perpendicular to the line $3x + y = 3$. Its y-intercept is
1. $\dfrac{1}{3}$
2. $\dfrac{2}{3}$
3. $1$
4. $\dfrac{4}{3}$

Answer 1

Hint: In this question, we are given an equation of line $3x + y = 3$ and we have to find the equation of the line perpendicular to $3x + y = 3$ and is passing through the point $\left( {2,2} \right)$. The first step is to compare the equation of the line with the general equation, and you’ll get the slope of the line. Now to calculate the slope of perpendicular line use ${m_1}{m_2} = - 1$. In last to find the equation apply $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$ where $\left( {{x_1},{y_1}} \right)$ are the points line is passing through. Now, compare it with $y = mx + c$. Here, $c$ is the y-intercept of the line.

Formula Used:
General equation of a straight line –
$y = mx + c$, where $m$ is the slope and $c$ is the y-intercept of the line.
When two lines are perpendicular to each other, and their slopes are ${m_1}$ and ${m_2}$ then ${m_1}{m_2} = - 1$
If a line is passing through a point $\left( {{x_1},{y_1}} \right)$ and its slope is $m$ then the equation of line is $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$

Complete step by step Solution:
Given that,
A line is passing through the point $\left( {2,2} \right)$ and perpendicular to the line $3x + y = 3$,
It can also be written as,
$y = - 3x + 3$-----(1)
Compare equation (1) with the general equation of a straight line $y = mx + c$,
It implies that, ${m_1} = - 3$
Therefore, slope of the equation $3x + y = 3$ is $ - 3$.
Now, the slope of perpendicular line is ${m_2} = - \dfrac{1}{{{m_1}}} = \dfrac{1}{3}$
Equation of the line passing through the point $\left( {2,2} \right)$ and the slope is $\dfrac{1}{3}$,
$\left( {y - {y_1}} \right) = {m_2}\left( {x - {x_1}} \right)$, Here ${x_1} = 2,{x_2} = 2$
$\therefore \left( {y - 2} \right) = \dfrac{1}{3}\left( {x - 2} \right)$
$ \Rightarrow 3y - 6 = x - 2$
$3y = x + 4$
$y = \dfrac{1}{3}x + \dfrac{4}{3}$
Compare the above equation with the general equation of a straight line $y = mx + c$,
$ \Rightarrow c = \dfrac{4}{3}$
Therefore, the y-intercept of the line perpendicular to $3x + y = 3$ is $\dfrac{4}{3}$.

Hence, the correct option is 4.

Note: The key concept involved in solving this problem is a good knowledge of the Equation of a line. Students must know that a line equation is easily understood as a single representation for multiple points on the same line. A line's equation has a general form, which is \[ax + by + c = 0\], and any point on this line satisfies this equation. The slope of the line and any point on the line are two absolutely necessary requirements for forming the equation of a line.