The coordinate axes are rotated about the origin in the counter-clockwise direction through an angle $60^{\circ}$. If $a$ and $b$ are the intercepts made on the new axes by a straight line whose equation referred to the original axes is $x+y=1$,then $\frac{1}{a^2}+\frac{1}{b^2}=$

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

When the origin is shifted to $(-1, 2)$ by the translation of axes,the transformed equation of $x^2+y^2+2x-4y+1=0$ is

Find the transformed equation of $x \cos \theta + y \sin \theta = p$,when the axes are rotated through an angle $\theta$.

By shifting the origin to the point $(2,3)$ through translation of axes,if the equation of the curve $x^2+3xy-2y^2+4x-y-20=0$ is transformed to the form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$,then $D+E+F=$