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:

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 $(-5, 3)$?

Let $L$ be the line $2x + y = 2$. If the axes are rotated by $45^\circ$,then the intercepts made by the line $L$ on the new axes are respectively

The transformed equation of $x^2-y^2+2x+4y=0$ when the origin is shifted to the point $(-1, 2)$ is

The point $P(\alpha, \beta)$ with $\alpha > 0, \beta > 0$ undergoes the following transformations successively:
$a)$ Translation by $3$ units in the positive direction of the $x$-axis.
$b)$ Reflection about the line $y = -x$.
$c)$ Rotation of axes through an angle of $\frac{\pi}{4}$ about the origin in the positive direction.
If the final position of the point $P$ is $(-4\sqrt{2}, -2\sqrt{2})$,then find the value of $(\alpha + \beta)$.

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