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

The slope of a line $L$ passing through the point $(-2,-3)$ is not defined. If the angle between the lines $L$ and $a x-2 y+3=0(a>0)$ is $45^{\circ}$, then the angle made by the line $x+a y-4 \doteq 0$ with positive $X$-axis in the anti-clockwise direction is

$(a, b)$ is the point of concurrency of the lines $x-3 y+3=0, k x+y+k=0$ and $2 x+y-8=0$. If the perpendicular distance from the origin to the line $L=a x-b y+2 k=0$ is $p$, then the perpendicular distance from the point $(2,3)$ to $L=0$ is

If $(4,3)$ and $(1,-2)$ are the end points of a diagonal of a square, then the equation of one of its sides is