Find the transformed equation of the curve $x^2+2 \sqrt{3} xy - y^2 = 8$,when the axes are rotated through an angle $\frac{\pi}{3}$.

By rotating the axes about the origin in an anti-clockwise direction by a certain angle,if the equation $x^2+4xy+y^2=1$ is transformed to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,then find the value of $\sqrt{\frac{a^2+b^2}{a^2}}$.

The angle through which the coordinate axes are to be rotated to remove the $xy$ term in the equation $x^2+2xy-y^2=0$ is

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

By shifting the origin to the point $(2, 3)$ and then rotating the coordinate axes through an angle $\theta$ in the counter-clockwise direction,if the equation $3x^2 + 2xy + 3y^2 - 18x - 22y + 50 = 0$ is transformed to $4X^2 + 2Y^2 - 1 = 0$,then the angle $\theta =$

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