When the axes are rotated through an angle $45^{\circ}$,the new coordinates of a point $P$ are $(1, -1)$. The coordinates of $P$ in the original system are

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

When the origin is shifted to $(1, -2)$ by translation of coordinate axes,the transformed coordinates of $(3, -2)$ are $(\alpha, \beta)$. If the axes are rotated about the origin through an angle of $45^{\circ}$ after the translation,then the transformed coordinates of $(\alpha, \beta)$ are

If the axes are transformed to the point $(-1, 1)$,then the equation $3x^2 + y^2 + 2x + 4y + 15 = 0$ would transform to:

If the origin is shifted to the point $(-2, 1)$,what are the new coordinates of the point $(4, -5)$?

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