Suppose the axes $X$ and $Y$ are obtained by rotating the axes $x$ and $y$ by an angle $\theta$. If the equation $x^2+2 \sqrt{3} x y-y^2=4 a^2$ is transformed to $X^2-Y^2=2 a^2$ with respect to the $XY$-axes,then $\theta$ is equal to (in $^{\circ}$)

The transformed equation of $3x^2 - 4xy = r^2$ when the coordinate axes are rotated through an angle $\tan^{-1}(2)$ is:

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 $2H(G+F)=$

When the coordinate axes are rotated through an angle $\theta$ in anti-clockwise direction,if the transformed equation of $x^2+y^2+2xy+2x+6y+1=0$ is $(2+\sqrt{3})X^2+2XY+(2-\sqrt{3})Y^2+aX+bY+2=0$,then $3a-b=$

If the axes are transformed to the point $(-1, 1)$,then the equation $3x^2 + y^2 + 2x + 4y + 15 = 0$ would transform to:

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