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

When the equation $x^2 - 3xy + 11x - 12y + 36 = 0$ is transformed by shifting the origin to the point $(-4, 1)$ while keeping the axes parallel,it becomes $ax^2 + bxy + 1 = 0$. Then $b^2 - a = \dots$

When the coordinate axes are rotated through an angle $\frac{\pi}{4}$ in the positive direction,an equation is transformed to $x^2+y^2-6x+8y+21=0$. Then the original equation is

If the origin is shifted to a point $P$ by the translation of axes to remove the $y$-term from the equation $x^2-y^2+2y-1=0$,then the transformed equation is:

$A$ line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle $\theta$ keeping the origin fixed,this line $L$ has the intercepts $p$ and $q$. Then

If the origin of a coordinate system is shifted to $(-\sqrt{2}, \sqrt{2})$ and the coordinate system is rotated anti-clockwise through an angle $45^{\circ}$,then the point $P(1, -1)$ in the original system has new coordinates