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

The point $P(4,1)$ undergoes the following transformations in succession:
$(i)$ origin is shifted to the point $(1,6)$ by translation of axes
(ii) translation through a distance of $2$ units along the positive direction of $X$-axis
(iii) rotation of axes through an angle of $90^{\circ}$ in the positive direction
Then the coordinates of the point $P$ in its final position are

When the origin is shifted to $(h, k)$ by translation of axes,the transformed equation of $x^2+2x+2y-7=0$ does not contain the $x$ term and the constant term. Then $(2h+k) =$

Statement $(A) :$ The area of the triangle formed by the points $A (20, 22), B (21, 24),$ and $C (22, 23)$ is equal to the area of the triangle formed by the points $P (0, 0), Q (1, 2),$ and $R (2, 1).$
Reason $(R) :$ The area of a triangle remains invariant under the translation of axes.

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 $(1, 1)$?

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