Question

If the axes are rotated through an angle ${30}^o$, then the coordinates of $(2\sqrt{3},-3)$ in the new system are

• A. $\left(\frac{3}{2},-\frac{5}{2}\right)$
• B. $\left(-\frac{\sqrt{3}}{2},\frac{5}{2}\right)$
• C. $\left(\frac{3}{2},-\frac{5\sqrt{3}}{2}\right)$
• D. $\left(3\sqrt{2},-\frac{5\sqrt{3}}{2}\right)$

Answer 1

Answer: (C)

$\left(\frac{3}{2},-\frac{5\sqrt{3}}{2}\right)$

By rotation, through an angle $\theta$
$x^{\prime }=x\cos \theta +y\sin \theta$
$y^{\prime }=-x\sin \theta +y\cos \theta$
$x^{\prime }=2\sqrt{3}\times \frac{\sqrt{3}}{2}-3\times \frac{1}{2}$

$=3-\frac{3}{2}=\frac{3}{2}$

$y^{\prime }=-2\sqrt{3}\times \frac{1}{2}-3\times \frac{\sqrt{3}}{2}$
$=\frac{-5\sqrt{3}}{2}$

$\therefore$ new co-ordinates are $(\frac{3}{2},-\frac{5\sqrt{3}}{2})$

Hence, option C is correct.