$A$ straight line is parallel to the lines $3x - y - 3 = 0$ and $3x - y + 5 = 0$ and lies between them. If its distance from these lines is in the ratio $3 : 5$,then its equation is:

The vertices of a triangle $OBC$ are $(0,0)$,$(-3,-1)$,and $(-1,-3)$ respectively. The equation of the line parallel to $BC$ which is at a distance of $\frac{1}{2}$ unit from the origin and cuts $OB$ and $OC$ is:

The points $(2,3)$ and $(-4,-4/3)$ lie on the opposite sides of the line $L \equiv 5x - 6y + k = 0$,where $k$ is an integer. If the points $(1,2)$ and $(4,5)$ lie on the same side of the line $L = 0$,then the perpendicular distance from the origin to the line $L = 0$ is:

The vertex of an equilateral triangle is $(2, -1)$ and the equation of its base is $x + 2y - 1 = 0$. The length of its side is

If the length of the perpendicular drawn from the point $(4,1)$ to the line $3x - 4y + k = 0$ is $2$ units,then the values of $k$ are:

If the point $(1, a)$ lies between the straight lines $x + y = 1$ and $2(x + y) = 3$,then $a$ lies in the interval