Question

The line joining two points A (2, 0) and B (3, 1) is rotated about A in anti-clockwise direction through an angle of \[{{15}^{\circ }}\] . The equation of the line in the now position, is

a)\[\sqrt{3}x-y-2\sqrt{3}=0\]
b)\[x-3\sqrt{y}-2=0\]
c)\[\sqrt{3}x+y-2\sqrt{3}=0\]
d)\[x+\sqrt{3}y-2=0\]

Answer 1

HINT: -The formula to calculate the slope of a line between two points is given as follows
\[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]

Complete step-by-step answer:

Another formula to calculate the slope of a line is given as follows

\[m=\tan \theta \]

(Where \[\theta \] is the angle made by the line with the x-axis of which the slope is being calculated)

As the line is rotated anti-clockwise keeping the point A as fixed, so the slope of the new line is increased by \[{{15}^{\circ }}\].


As mentioned in the question, we have to find the equation of the new line that is formed when the original line is rotated anti-clockwise keeping the point A fixed.

Now, using the formula given in the hint, we can find the slope of the line as follows
\[m=\dfrac{1-0}{3-2}=1\]

Hence, the angle made by this line with the x-axis is given as follows

\[\begin{align}

  & \tan \theta =1 \\

 & \theta ={{45}^{\circ }} \\

\end{align}\]

Now, the line is rotated by \[{{15}^{\circ }}\] , so the slope of the new line obtained is as follows

\[\begin{align}

  & m=\tan \left( {{45}^{\circ }}+{{15}^{\circ }} \right) \\

 & m=\tan ({{60}^{\circ }}) \\

 & m=\sqrt{3} \\

\end{align}\]


Hence, the equation of the new line can be written as follows

\[~y=\sqrt{3}x+c\]

But it passes through (2, 0) that is point A, so, we can write as follows

\[\begin{align}

  & ~y=\sqrt{3}x+c \\

 & 0=2\sqrt{3}+c \\

 & c=-2\sqrt{3} \\

\end{align}\]

Therefore, the equation of the new line is as follows

\[\sqrt{3}x-y-2\sqrt{3}=0\]


NOTE: -The students can make an error if they don’t know the general properties of the equation of a line which are mentioned in the hint as follows

The formula to calculate the slope of a line between two points is given as follows

\[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]

Another formula to calculate the slope of a line is given as follows

\[m=\tan \theta \]

(Where \[\theta \] is the angle made by the line with the x-axis of which the slope is being calculated)