Question

If the coordinate of a point $P$ are transformed to $(4,-6\sqrt{3})$ when the axis are rotated through an angle $30{}^{\circ }$ in the anti clockwise direction, then original coordinates of $P$ are

• A. $(\sqrt{3},4)$
• B. $(5\sqrt{3},7)$
• C. $(5\sqrt{3},-7)$
• D. $(4\sqrt{3},-7)$

Answer 1

The correct option is C $(5\sqrt{3},-7)$

$(X,Y)=(4,-6\sqrt{3}),\theta =30{}^{\circ }x=X\cos \theta -Y\sin \theta =4\left(\frac{\sqrt{3}}{2}\right)+6\sqrt{3}\left(\frac{1}{2}\right)=5\sqrt{3}y=X\sin \theta +Y\cos \theta =4\left(\frac{1}{2}\right)-6\sqrt{3}\left(\frac{\sqrt{3}}{2}\right)=-7$
Hence original coordinate of $P=(5\sqrt{3},-7)$