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 $\alpha=$

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

The angle through which the coordinate axes are to be rotated to remove the $xy$ term in the equation $x^2+2xy-y^2=0$ is

The mixed term $xy$ is to be removed from the general equation $ax^2 + by^2 + 2hxy + 2fy + 2gx + c = 0$. One should rotate the axes through an angle $\theta$ given by $\tan 2\theta$ equal to:

The point $P(3,2)$ undergoes the following transformations successively:
$(i)$ Reflection about the line $y=x$
(ii) Translation to a distance of $3$ units in the positive direction of $X$-axis
(iii) Rotation through an angle $\frac{\pi}{4}$ about the origin in the counter-clockwise direction
Then,the final position of that point is

The coordinate axes are rotated through an angle $135^{\circ}$. If the coordinates of a point $P$ in the new system are known to be $(4, -3)$,then the coordinates of $P$ in the original system are