When the origin is shifted to the point $(2, b)$ by translation of axes,the coordinates of the point $(a, 4)$ change to $(6, 8)$. When the origin is shifted to $(a, b)$ by translation of axes,if the transformed equation of $x^2+4xy+y^2=0$ is $X^2+2HXY+Y^2+2GX+2FY+C=0$,then $2H(G+F)=$

Suppose the axes $X$ and $Y$ are obtained by rotating the axes $x$ and $y$ by an angle $\theta$. If the equation $x^2+2 \sqrt{3} x y-y^2=4 a^2$ is transformed to $X^2-Y^2=2 a^2$ with respect to the $XY$-axes,then $\theta$ is equal to (in $^{\circ}$)

If the origin is shifted to a point $(h, k)$ by translation of axes in order to make the equation $x^2+5xy+2y^2+5x+6y+7=0$ free from first-order terms,then:

The equation of a curve $C$ is transformed to $X^2+Y^2-6X+8Y+21=0$ by the rotation of coordinate axes about the origin through an angle of $\frac{\pi}{4}$ in the positive direction. If $ax^2+by^2+cx+dy+e=0$ is the equation of the curve $C$ before the transformation,then find the value of $(a+b+c^2+d^2-5e)^2$.

The transformed equation of $x^2+y^2=r^2$ when the axes are rotated through an angle $36^{\circ}$ is

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