$(a, b)$ is the point to which the origin has to be shifted by translation of axes so as to remove the first-degree terms from the equation $2x^2 - 3xy + 4y^2 + 5y - 6 = 0$. If the angle by which the axes are to be rotated in the positive direction about the origin to remove the $xy$-term from the equation $ax^2 + 23abxy + by^2 = 0$ is $\theta$,then $\tan 2\theta =$

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 axes are rotated by an angle of $-\pi /3$ in the negative direction and the coordinates of a point in the new system are $(4, 2)$,find the coordinates of the point in the original system.

The equation of a curve $C$ is transformed to $X^2+Y^2-6X+8Y+21=0$ by the rotation of coordinate axes about the origin through an angle of $\frac{\pi}{4}$ in the positive direction. If $ax^2+by^2+cx+dy+e=0$ is the equation of the curve $C$ before the transformation,then find the value of $(a+b+c^2+d^2-5e)^2$.

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 $(\alpha + \beta)$.

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