If the axes are rotated through an angle $45^{\circ}$ in the positive direction without changing the origin,then the coordinates of the point $(\sqrt{2}, 4)$ in the old system are

Find the new coordinates of the reflection of the point $(2, -1)$ about the $y$-axis,when the axes are rotated by $45^{\circ}$ in the negative direction without shifting the origin.

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

If the origin is shifted to the point $(2, -5)$ and the axes are kept parallel,what will be the new coordinates of the point $(-5, 3)$?

The point $P(1,4)$ occupies the positions $A, B$ and $C$ respectively after undergoing the following three transformations successively:
$I$. Reflection about the line $y=x$.
$II$. Translation through a distance of $1$ unit along the positive direction of $X$-axis.
$III$. Rotation of the line $OB$ through an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction. Then,the coordinates of $C$ are

Suppose the axes are to be rotated through an angle $\theta$ so as to remove the $xy$ term from the equation $3x^2+2\sqrt{3}xy+y^2=0$. Then in the new coordinate system,the equation $x^2+y^2+2xy=2$ is transformed to: