The distance between the lines given by $3x + 4y = 9$ and $6x + 8y = 15$ is (in $units$)

If $P(\alpha, \beta)$ is a point on the line $3x + y = 0$ such that the point $P$ and the point $Q(1, 1)$ lie on opposite sides of the line $3x - 4y - 8 = 0$,then:

The number of lines that can be drawn through the point $(4, -5)$ at a distance of $10$ units from the point $(1, 3)$ is

Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $(k, \frac{k}{2})$ from the line $3x+4y+5=0$ is

If $2p$ is the length of the perpendicular from the origin to the line $\frac{x}{a} + \frac{y}{b} = 1$,then $a^2, 8p^2, b^2$ are in

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: