If the coordinates of a point $P$ change to $(2, -6)$ when the coordinate axes are rotated through an angle of $135^{\circ}$,then 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

The point to which the origin should be shifted in order to eliminate the $x$ and $y$ terms from the equation $9x^2+4y^2+10x+12y+1=0$ is

$A$ line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle $\theta$ keeping the origin fixed,this line $L$ has the intercepts $p$ and $q$. Then

The point $(4, 1)$ undergoes the following three transformations successively: $(i)$ Reflection about the line $y = x$,(ii) Translation through a distance of $2$ units along the positive direction of the $x$-axis,(iii) Rotation through an angle $\pi/4$ about the origin in the anti-clockwise direction. The final position of the point is given by the coordinates:

If the origin of a coordinate system is shifted to $(-\sqrt{2}, \sqrt{2})$ and the coordinate system is rotated anti-clockwise through an angle $45^{\circ}$,then the point $P(1, -1)$ in the original system has new coordinates