Question

What is the angle formed between the x-axis and the line passes through the points \[\left( {1,0} \right)\] and \[\left( { - 2,\sqrt 3 } \right)\]?
A. \[60^{\circ}\]
B. \[120^{\circ}\]
C. \[150^{\circ}\]
D. \[135^{\circ}\]

Answer 1

Hint: First, use the formula of slope to find the slope of the line passing through the given points. Then use the formula of the angle between a line and the x-axis to reach the required answer.

Formula Used:
Slope formula: The slope of the line passing through the points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] is: \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
The angle between a line with slope \[m\] and the x-axis is: \[\theta = \tan^{ - 1}\left( m \right)\]
\[\tan^{ - 1}\left( {\tan A} \right) = A\]

Complete step by step solution:
Given:
The line passes through the points \[\left( {1,0} \right)\] and \[\left( { - 2,\sqrt 3 } \right)\].
Let’s calculate the slope of the line that passes through the given points.
Let \[m\] be the slope of the line.
Apply the slope formula.
\[m = \dfrac{{\sqrt 3 - 0}}{{ - 2 - 1}}\]
\[ \Rightarrow \]\[m = \dfrac{{\sqrt 3 }}{{ - 3}}\]
\[ \Rightarrow \]\[m = \dfrac{- 1}{{ \sqrt 3 }}\]
Now we have to calculate the angle between the x-axis and the line passing through the points \[\left( {1,0} \right)\] and \[\left( { - 2,\sqrt 3 } \right)\].
Let \[a\] be the angle between the line and the x-axis.
Apply the formula of the angle between the line and the x-axis.
\[a = \tan^{ - 1}\left( m \right)\]
Substitute the value of slope in the above equation.
\[a = \tan^{ - 1}\left( {\dfrac{{ - 1}}{{\sqrt 3 }}} \right)\]
\[ \Rightarrow \]\[a = \tan^{ - 1}\left( {\tan\left( {150^{\circ}} \right)} \right)\]
\[ \Rightarrow \]\[a = 150^{\circ}\] [Since \[\tan^{ - 1}\left( {\tan A} \right) = A\]]

Hence the correct option is C.

Note: Students often get confused about the formula of the angle between the line and the x-axis. When the slope of the line is known, the angle between the line and the x-axis can be calculated.