The straight lines joining the origin to the points of intersection of the line $2x + y = 1$ and the curve $3x^2 + 4xy - 4x + 1 = 0$ include an angle of:

The equations of the lines which pass through the origin and are inclined at an angle $\tan^{-1} m$ to the line $y = mx + c$ are

Consider the lines $L_1 \equiv 4x + 5y - 6 = 0$,$L_2 \equiv 2x + 3y - 4 = 0$,and $L_3 \equiv 3x - y + 2 = 0$. If the line $L_1 = 0$ intersects the lines $L_2 = 0$ and $L_3 = 0$ at the points $A$ and $B$ respectively,then the combined equation of lines $OA$ and $OB$ is

The line $x + y = 3$ intersects the pair of straight lines $x^2 - 3xy + y^2 = 0$ at points $A$ and $B$. Then the coordinates of the mid-point of $AB$ are

The distance between the parallel lines given by $(x+7y)^2 + 4\sqrt{2}(x+7y) - 42 = 0$ is

The pair of lines joining the origin to the points of intersection of the line $x-y=2$ with the curve $5x^2+12xy-8y^2+8x-4y+12=0$ are equally inclined to the pair of lines