The combined equation of the bisectors of the angles between the lines joining the origin to the points of intersection of the curve $x^2+y^2+xy+x+3y+1=0$ and the line $x+y+2=0$ is

The distance between the pair of parallel lines represented by $x^2+2xy+y^2-8ax-8ay-9a^2=0$ is $...$ units.

If $f(x, y) = 0$ is the combined equation of the lines joining the origin to the points where the line $4x - 6y - 2 = 0$ meets the curve $3x^2 - 4xy + 5y^2 - 2x + y - 6 = 0$,then $\frac{f(1, -1)}{f(-1, -1)} = $

Let $3x^2+8xy-3y^2=0$ represent the lines $L_1, L_2$ and $3x^2+8xy-3y^2+2x-4y-1=0$ represent the lines $L_3, L_4$. Let $L$ be the line joining the points of intersection of $L_1, L_3$ and $L_2, L_4$. Then,the area (in sq units) of the triangle formed by $L$ with the coordinate axes is

The distance between the lines represented by $16x^2 - 24xy + 9y^2 + 48x - 36y + 35 = 0$ is ...... units.

The line $x+2y=k$ meets the curve $2x^2-2xy+3y^2+2x-y-1=0$ at two points $A$ and $B$. Let $O$ be the origin. If the line segments $OA$ and $OB$ are perpendicular to each other,then $k=$