Question

If the coordinates of the points $A$ and $B$ be $(1,0)$ and $(2,\sqrt{3})$, then the angle made by the line $\overleftrightarrow{AB}$ with the x-axis is
A. $30{}^\circ $
B. $45{}^\circ $
C. $60{}^\circ $
D. $75{}^\circ $

Answer 1

Hint: In this question, we are to find the angle made by the line with the x-axis. Since we have the coordinates of the line, we can able to find the slope. By this, we can find the angle of the line with the x-axis.



Formula Used:The slope of a line is
$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
The angle made by the line with the x-axis in the positive direction is given by the slope of the line as
$m=\tan \theta $



Complete step by step solution:Given that,
The ends of a line $\overleftrightarrow{AB}$ are $A(1,0)$ and $B(2,\sqrt{3})$
Then, its slope is
$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
On substituting,
$\begin{align}
  & m=\dfrac{\sqrt{3}-0}{2-1} \\
 & \text{ }=\sqrt{3} \\
\end{align}$
The angle made by the line $\overleftrightarrow{AB}$ with the x-axis is
$\begin{align}
  & m=\tan \theta \\
 & \Rightarrow \sqrt{3}=\tan \theta \\
 & \Rightarrow \theta ={{\tan }^{-1}}\left( \sqrt{3} \right) \\
 & \therefore \theta =60{}^\circ \\
\end{align}$



Option ‘C’ is correct



Note: Here we need to remember that the slope of a line is formed by the inclination of the line with the x-axis in the positive direction. Thus, the angle of the inclination will be the required angle made by the line. So, in order to find the angle made by the line, the slope of the line is required.