Question

Find equation of the line through the point \[\left( {0,2} \right)\] making an angle \[\dfrac{{2\pi }}{3}\] with the positive \[x\] -axis. Also, find the equation of line parallel to it and crossing the y-axis at a distance of 2 units below the origin.

Answer 1

Hint:To solve the question given above, we will use the formula for finding the slope of the line. After finding the slope we will use the standard formula for the equation of the line. To solve this question, we also need to keep in mind the concept of parallel lines and the relationship between their slopes.

Formula used: The formulas that we will be using the solve the above question are:
To find the slope of a line: \[m = \tan \theta \] , here \[m\] refers to the slope of the line.
The equation of a line is: \[y - {y_1} = m\left( {x - {x_1}} \right)\]

Complete step by step solution:
First, we will find the slope of the line making an angle \[\dfrac{{2\pi }}{3}\] with the positive \[x\] -axis.
For this, we will use the 1) formula, \[m = \tan \theta \] now we are given \[\theta = \dfrac{{2\pi }}{3}\] .
So, \[m = \tan \dfrac{{2\pi }}{3}\]
\[ = - \sqrt 3 \] .
Now, we will find the equation of the line through the point \[\left( {0,2} \right)\] with the slope \[ - \sqrt 3 \] .
Now we will be using 2) formula, \[y - {y_1} = m\left( {x - {x_1}} \right)\] , we get,
\[
  y - 2 = - \sqrt 3 \left( {x - 0} \right) \\
   \Rightarrow y - 2 = - \sqrt 3 x \\
   \Rightarrow \sqrt 3 x + y - 2 = 0 \\
 \] .
After this, we will find the slope of the line parallel to line \[\sqrt 3 x + y - 2 = 0\].
We know that the parallel lines have the same slope, therefore, the line parallel to \[\sqrt 3 x + y - 2 = 0\] has slope \[ - \sqrt 3 \] .
We are given that the equation of line parallel to \[\sqrt 3 x + y - 2 = 0\] crosses the origin at the y-axis at a distance of 2 units below the origin. So, it passes through the point \[\left( {0, - 2} \right)\] .
So, the equation of line passing through \[\left( {0, - 2} \right)\] , having a slope \[ - \sqrt 3 \] is:
\[
  y - \left( { - 2} \right) = - \sqrt 3 \left( {x - 0} \right) \\
   \Rightarrow y + 2 = - \sqrt 3 x \\
 \]
This can be written as:
\[\sqrt 3 x + y + 2 = 0\] .

Hence, our final answer is: \[\sqrt 3 x + y + 2 = 0\] .

Note: While solving questions similar to one given above, you need to keep in mind the formulas used for finding the slope of a line and the equation of the line. Also, you need to be familiar with the concepts of parallel lines and perpendicular lines. We know that the parallel lines have the same slopes whereas in case of perpendicular lines the slope of the second line is equal to the negative reciprocal of the slope of the first line.