Question

A line joining two points $A(2,0)$ and $B(3,1)$ is rotated about $A$ in anti-clockwise direction through an angle ${15}^0$. If goes to $C$ in the new position, then the coordinates of $C$ are

• A. $(2,\sqrt{\frac{3}{2}})$
• B. $(2,\sqrt{-\frac{3}{2}})$
• C. $(2+\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}})$
• D. None of these

Answer 1

The correct option is C $(2+\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}})$

(c) Slope of line $AB=\frac{0-1}{2-3}=1=\tan {45}^0$

$\therefore \angle BAX={45}^0$

Given $\angle CAB={15}^0$

$\therefore \angle CAX={60}^0$

$\therefore$ Slope of line $AC=\tan {60}^0=\surd 3$

Now line $AC$ makes an angle of ${60}^0$ with positive direction of$x$-axis and

$AC=AB=\sqrt{{(3-2)}^2+{(1-0)}^2}=\sqrt{2}$

$\therefore$ Co-ordinates of $C$ are $(2+\sqrt{2}\cos {60}^0,0+\sqrt{2}\sin {60}^0)$

i.e.,$(2+\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}})$