Question

A line passes through point $(2,2)$ and perpendicular to the line $3x+y=3$. Then, $y-$intercept is

• A. $2$
• B. $\frac{5}{3}$
• C. $\frac{4}{3}$
• D. $\frac{3}{4}$

Answer 1

The correct option is C

$\frac{4}{3}$

Step1. Finding the slope of other line

Given line, $3x+y=3$

Therefore, $y=-3x+3$

So, the slope the given line is ${\mathrm{m}}_1=-3$
For two lines to be mutually perpendicular to each other,

Product of their slope is ${\mathbf{m}}_{\mathbf{1}}\mathbf{\times }{\mathbf{m}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{1}$

Therefore, ${\mathrm{m}}_2=\frac{1}{3}$

Step2. Finding the equation of line

Now, equation line passing through the point $(2,2)$ and slope ${\mathrm{m}}_2=\frac{1}{3}$ is

$$\begin{gathered} \Rightarrow \mathrm{y}-2=\frac{1}{3}(\mathrm{x}-2)\mathbf{y}\mathbf{-}{\mathbf{y}}_{\mathbf{1}}\mathbf{=}{\mathbf{m}}_{\mathbf{2}}\mathbf{(}\mathbf{x}\mathbf{-}{\mathbf{x}}_{\mathbf{2}}\mathbf{)} \\ \Rightarrow \mathrm{y}=\frac{\mathrm{x}}{3}+\frac{4}{3} \end{gathered}$$

For $y-$intercept put $\mathrm{x}=0$

Therefore, $y-$intercept is $\frac{4}{3}$

Hence, the correct option is $\mathbf{\left(}\mathit{C}\mathbf{\right)}$.