When the origin is shifted to the point $(2,3)$ and then the coordinate axes are rotated through an angle $\frac{\pi}{3}$ in the counter-clockwise sense,then the transformed equation of $3 x^2+2 x y+3 y^2-18 x-22 y+50=0$ is

The transformed equation of $x^2+6xy+8y^2=10$ when the axes are rotated through an angle $\frac{\pi}{4}$ is:

Point $(-1, 2)$ is changed to $(a, b)$ when the origin is shifted to the point $(2, -1)$ by translation of axes. Point $(a, b)$ is changed to $(c, d)$ when the axes are rotated through an angle of $45^{\circ}$ about the new origin. Point $(c, d)$ is changed to $(e, f)$ when $(c, d)$ is reflected through the line $y = x$. Then $(e, f) =$

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

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

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