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

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 point to which the origin is to be shifted by translation of axes so that the transformed equation of $y^2+4y+8x-2=0$ will not contain $y$ term and constant term is

By rotating the axes through an angle of $30^{\circ}$ in the anti-clockwise direction about the origin,the equation $4x^2+12xy+9y^2+6x+9y+2=0$ becomes $ax^2+2hxy+by^2+2gx+2fy+c=0$. Then which of the following is true?

If $(h, k)$ is the new origin to be chosen to eliminate first degree terms from the equation $S \equiv 2x^2 - xy - y^2 - 3x + 3y = 0$ by translation and if $\theta$ is the angle with which the axes are to be rotated about the origin in anticlockwise direction to eliminate the $xy$-term from $S = 0$,then $\tan 2\theta =$

The point $P(3,2)$ undergoes the following transformations successively:
$(i)$ Reflection about the line $y=x$
(ii) Translation to a distance of $3$ units in the positive direction of $X$-axis
(iii) Rotation through an angle $\frac{\pi}{4}$ about the origin in the counter-clockwise direction
Then,the final position of that point is