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

Transforming to parallel axes through a point $(p, q)$, the equation $2x^2 + 3xy + 4y^2 + x + 18y + 25 = 0$ becomes $2x^2 + 3xy + 4y^2 = 1$. Then:

If the axes are rotated through an angle $45^{\circ}$,then the coordinates of the point $(4 \sqrt{2}, -6 \sqrt{2})$ in the new system are . . . . . .

If the origin is shifted to $(1, -2)$ and the axes are rotated by an angle of $30^{\circ}$,what will be the new coordinates of $(1, 1)$?

Let $P$ be the point to which the origin is shifted by the translation of axes so as to remove the first-degree terms from the equation $3x^2+y^2-6x+4y+4=0$. If the origin is shifted to $P$ by the translation of axes,then the transformed equation of $2x^2+3xy-5y^2+2x-23y-24=0$ is

The transformed equation of $3x^2 - 4xy = r^2$ when the coordinate axes are rotated through an angle $\tan^{-1}(2)$ is: