Question

The line joining two points $A(2,0),B(2+\sqrt{3},1)$ is rotated about $A$ in anti-clockwise direction through an angle of ${30}^{\circ }$. Then the equation of new line is

• A. $\frac{x+2}{1/2}=\frac{y}{\sqrt{3}/2}$
• B. $\frac{x-2}{1/2}=\frac{y}{\sqrt{3}/2}$
• C. $\frac{x}{1/2}=\frac{y}{\sqrt{3}/2}$
• D. None of these

Answer 1

The correct option is B $\frac{x-2}{1/2}=\frac{y}{\sqrt{3}/2}$

The slope of the line $AB$ is given by
$m=\frac{1-0}{2+\sqrt{3}-2}=\frac{1}{\sqrt{3}}$
$\Rightarrow \frac{1}{\sqrt{3}}=\tan \theta \Rightarrow \theta =\frac{\pi }{6}$
$\therefore$ The line makes on angle of ${30}^{\circ }$ with $x-$axis.
$AB$ is rotated through ${30}^{\circ }$about point $A$ in anticlock wise. So, it makes an angle of ${60}^{\circ }$ with $x-$axis. Clearly, line still passes through $A(2,0)$.
Therefore, the equation of $AC$ is
$\frac{x-2}{\cos {60}^{\circ }}=\frac{y-0}{\sin {60}^{\circ }}$
$\Rightarrow \frac{x-2}{1/2}=\frac{y}{\sqrt{3}/2}$