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

The relation between coordinates $(x,y)and(x^{\prime }.y^{\prime })$ can be expressed as

$x^{\prime }=x\cos \theta +y\sin \theta$

$y^{\prime }=-x\sin \theta +y\cos \theta$

$x^{\prime }=\left(4\right)\cos {30}^o+(-2\sqrt{3})\sin {30}^o$

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

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

$=\sqrt{3}$

$y^{\prime }=-\left(4\right)\sin \left({30}^o\right)+(-2\sqrt{3})\cos \left({30}^o\right)$

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

$=-5$

$\therefore$ $(x^{\prime },y^{\prime })=(\sqrt{3},-5)$

$\therefore$ The required co-ordinates are $(\sqrt{3},-5)$