The point to which the origin should be shifted in order to eliminate the $x$ and $y$ terms from the equation $9x^2+4y^2+10x+12y+1=0$ is

The transformed equation of $3x^2 + 3y^2 + 2xy = 2$,when the coordinate axes are rotated through an angle of $45^{\circ}$,is

If the area of the region bounded by the curves $y=x^2$ and $x=y^2$ is $k$,then the area of the region bounded by the curves $\frac{x+\sqrt{3} y}{2}=\left(\frac{\sqrt{3} x-y}{2}\right)^2$ and $\frac{\sqrt{3} x-y}{2}=\left(\frac{x+\sqrt{3} y}{2}\right)^2$ is:

The transformed equation of $3x^2 - 4xy = r^2$ when the coordinate axes are rotated about the origin through an angle of $\operatorname{Tan}^{-1}(2)$ in the positive direction is:

If $(a, b)$ are the new coordinates of the point $(2, 3)$ after shifting the origin to the point $(3, 2)$ by translation of axes,and $(c, d)$ are the new coordinates of the point $(a, b)$ after rotating the axes through an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction,then find the value of $d-c$.

If $\theta_1, \theta_2, \theta_3$ are respectively the angles by which the coordinate axes are to be rotated to eliminate the $xy$ term from the following equations,then the descending order of these angles is:
$A_1 = 3x^2 + 5xy + 3y^2 + 2x + 3y + 4 = 0$
$A_2 = 5x^2 + 2\sqrt{3}xy + 3y^2 + 6 = 0$
$A_3 = 4x^2 + \sqrt{3}xy + 5y^2 - 4 = 0$