When the coordinate axes are rotated about the origin through an angle $\frac{\pi}{4}$ in the positive direction,the equation $ax^2+2hxy+by^2=c$ is transformed to $25x^2+9y^2=225$,then $(a+2h+b-\sqrt{c})^2=$

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 $b^2 - a = \dots$

The point $P(1,4)$ occupies the positions $A, B$ and $C$ respectively after undergoing the following three transformations successively:
$I$. Reflection about the line $y=x$.
$II$. Translation through a distance of $1$ unit along the positive direction of $X$-axis.
$III$. Rotation of the line $OB$ through an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction. Then,the coordinates of $C$ are

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi}{4}$,if the equation $25x^2+9y^2=225$ is transformed to $\alpha x^2+\beta xy+\gamma y^2=\delta$,then $(\alpha+\beta+\gamma-\sqrt{\delta})^2=$

The point to which the origin should be shifted so that the equation $y^2-6y-4x+13=0$ will not contain any term in $y$ and the constant term,is

After the coordinate axes are rotated through an angle $\frac{\pi}{4}$ in the anti-clockwise direction without shifting the origin,if the equation $x^2+y^2-2x-4y-20=0$ transforms to $ax^2+2hxy+by^2+2gx+2fy+c=0$ in the new coordinate system,then $\left|\begin{array}{lll}a & h & g \\ h & b & f \\ g & f & c\end{array}\right|=$