If the axes are rotated by an angle of $30^{\circ}$ in the counter-clockwise direction,find the coordinates of the point $(4, -2\sqrt{3})$ with respect to the new axes.

If the origin of a coordinate system is shifted to $(-\sqrt{2}, \sqrt{2})$ and the coordinate system is rotated anti-clockwise through an angle $45^{\circ}$,then the point $P(1, -1)$ in the original system has new coordinates

The transformed equation of $3x^2 - 4xy = r^2$ when the coordinate axes are rotated about the origin through an angle of $\operatorname{Tan}^{-1}(2)$ in the positive direction is:

The transformed equation of the curve $2x^2+y^2-3x+5y-8=0$ when the origin is translated to the point $(-1, 2)$ is

By rotating the coordinate axes in the positive direction about the origin by an angle $\alpha$,if the point $(1,2)$ is transformed to $\left(\frac{3 \sqrt{3}-1}{2 \sqrt{2}}, \frac{\sqrt{3}+3}{2 \sqrt{2}}\right)$ in the new coordinate system,then $\alpha=$

If the coordinate axes are rotated in the positive direction by $45^{\circ}$ without changing the origin,then the transformed equation of $3x^2 + 3y^2 + 2xy - 2 = 0$ is