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:

By shifting the origin to the point $(2,3)$ through translation of axes,if the equation of the curve $x^2+3xy-2y^2+4x-y-20=0$ is transformed to the form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$,then $D+E+F=$

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

When the coordinate axes are rotated through an angle $\frac{\pi}{4}$ in the positive direction,an equation is transformed to $x^2+y^2-6x+8y+21=0$. Then the original equation is

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

The point $P(\alpha, \beta)$ with $\alpha > 0, \beta > 0$ undergoes the following transformations successively:
$a)$ Translation by $3$ units in the positive direction of the $x$-axis.
$b)$ Reflection about the line $y = -x$.
$c)$ Rotation of axes through an angle of $\frac{\pi}{4}$ about the origin in the positive direction.
If the final position of the point $P$ is $(-4\sqrt{2}, -2\sqrt{2})$,then find the value of $(\alpha + \beta)$.