By rotating the axes through an angle of $30^{\circ}$ in the anti-clockwise direction about the origin,the equation $4x^2+12xy+9y^2+6x+9y+2=0$ becomes $ax^2+2hxy+by^2+2gx+2fy+c=0$. Then which of the following is true?

If the coordinate axes are rotated through an angle $\frac{\pi}{6}$ about the origin,then the transformed equation of $\sqrt{3} x^2-4 x y+\sqrt{3} y^2=0$ is

The point $P(\alpha, \beta)$ with $\alpha > 0, \beta > 0$ undergoes the following transformations successively:
$a)$ Translation by $3$ units in the positive direction of the $x$-axis.
$b)$ Reflection about the line $y = -x$.
$c)$ Rotation of axes through an angle of $\frac{\pi}{4}$ about the origin in the positive direction.
If the final position of the point $P$ is $(-4\sqrt{2}, -2\sqrt{2})$,then find the value of $(\alpha + \beta)$.

In order to eliminate the first degree terms from the equation $4x^2+8xy+10y^2-8x-44y+14=0$,the point to which the origin has to be shifted is

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

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