Question

If the axes are rotated through an angle of ${30}^o$ in the anti-clockwise direction, the coordinates of point $(4,-2\sqrt{3})$ with respect to new axes are-

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

Answer 1

The correct option is B $(\sqrt{3},-5)$

$x=r\cos \alpha$

$y=r\sin \alpha$

$x^1=r\cos (\alpha -{30}^o)$

$y^1=r\sin (\alpha -{30}^o)$

$x^1=r[\cos \alpha \cos {30}^o+\sin \alpha \sin {30}^o]$

$x^1=\frac{4\sqrt{3}}{2}-2\sqrt{3}\times \frac{1}{2}$

$x^1=2\sqrt{3}-\sqrt{3}$

$\therefore x^1=\sqrt{3}$

$y^1=r[\sin \alpha \cos {30}^o-\cos \alpha \sin {30}^o]$

$=-2\sqrt{3}\times \frac{\sqrt{3}}{2}-4\times \frac{1}{2}$

$y^1=-5$

$\therefore \alpha (\sqrt{3},-5)$