When the origin is shifted to the point $(2, b)$ by translation of axes,the coordinates of the point $(a, 4)$ change to $(6, 8)$. When the origin is shifted to $(a, b)$ by translation of axes,if the transformed equation of $x^2+4xy+y^2=0$ is $X^2+2HXY+Y^2+2GX+2FY+C=0$,then $2H(G+F)=$

The coordinates of the point $(3,-7,5)$ in the new system,when the origin is shifted to the point $(-1,-1,-1)$ by the translation of axes,are

If $(a, b)$ are the new coordinates of the point $(2, 3)$ after shifting the origin to the point $(3, 2)$ by translation of axes,and $(c, d)$ are the new coordinates of the point $(a, b)$ after rotating the axes through an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction,then find the value of $d-c$.

The transformed equation of $x^2+y^2=r^2$ when the axes are rotated through an angle $36^{\circ}$ is

If the point $P(1,3)$ undergoes the following transformations successively:
$(i)$ Reflection with respect to the line $y=x$.
(ii) Translation through $3$ units along the positive direction of the $X$-axis.
(iii) Rotation through an angle of $\frac{\pi}{6}$ about the origin in the clockwise direction.
Then,the final position of the point $P$ is

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 $\tan 2\theta =$