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 $c(a+b+d)=$

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 $^{\circ}$)

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

When the coordinate axes are rotated by an angle $\tan^{-1}\left(\frac{3}{4}\right)$ about the origin,then the equation $x^2+y^2=9$ is transformed to the equation

$A$ line $L$ makes intercepts $a$ and $b$ on the coordinate axes. The axes are rotated through an angle $\theta$ in the positive direction,keeping the origin fixed. If the line $L$ makes intercepts $p$ and $q$ on the new coordinate axes,then $\frac{1}{a^2}+\frac{1}{b^2}=$

The origin is translated to $(1,2)$. The point $(7,5)$ in the old system undergoes the following transformations successively.
$I$. Moves to the new point under the given translation of origin.
$II$. Translated through $2$ units along the negative direction of the new $X$-axis.
$III$. Rotated through an angle $\frac{\pi}{4}$ about the origin of the new system in the clockwise direction. The final position of the point $(7,5)$ is