If $a \alpha^2+b \beta^2+c \alpha \beta+d=0$ is the transformed equation of $4 x^2+\sqrt{3} x y+5 y^2-4=0$ obtained by using $\alpha=\frac{\sqrt{3}}{2} x+\frac{y}{2}$ and $\beta=-\frac{x}{2}+\frac{\sqrt{3}}{2} y$,then $c(a+b+d)=$

The point to which the origin should be shifted so that the equation $y^2-6y-4x+13=0$ will not contain any term in $y$ and the constant term,is

If the axes are rotated by an angle of $30^{\circ}$ in the counter-clockwise direction,find the coordinates of the point $(4, -2\sqrt{3})$ with respect to the new axes.

When the origin is shifted to $(1, -2)$ by translation of coordinate axes,the transformed coordinates of $(3, -2)$ are $(\alpha, \beta)$. If the axes are rotated about the origin through an angle of $45^{\circ}$ after the translation,then the transformed coordinates of $(\alpha, \beta)$ are

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 $\tan 2\theta =$

The point $P(1,4)$ occupies the positions $A, B$ and $C$ respectively after undergoing the following three transformations successively:
$I$. Reflection about the line $y=x$.
$II$. Translation through a distance of $1$ unit along the positive direction of $X$-axis.
$III$. Rotation of the line $OB$ through an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction. Then,the coordinates of $C$ are