Question

The line joining the points $A(2,0)$ and $B(3,1)$ is rotated through an angle of ${45}^o$, about $A$ in the anticlockwise direction. The coordinates of $B$ in the new position

• A. $(2,\sqrt{2})$
• B. $(\sqrt{2},2)$
• C. $(2,2)$
• D. $(\sqrt{2},\sqrt{2})$

Answer 1

The correct option is A $(2,\sqrt{2})$

Shift true origin to $A$. Then coordinate will be $A^{\prime }=(2-2,0)=(0,0)$ and $B^{\prime }=(3-2,1-0)=(1,1)$

Now, convert it to poles form,

$r=\sqrt{2}$ and $\varphi ={45}^o$

$\therefore B^{\prime }=(\sqrt{2}\cos {45}^o,\sqrt{2}\sin {45}^o)$

Now, rotate it by ${45}^o$.

So, new $B^{\prime \prime }=\left(\sqrt{2}\cos \right({45}^o+45),\sqrt{2}\sin ({45}^o+{45}^o)=(0,\sqrt{2})$

Now, convert it back to old coordinate $=(0+2,\sqrt{2})=(2,\sqrt{2})$