If the lines $x + 2ay + a = 0$,$x + 3by + b = 0$,and $x + 4cy + c = 0$ are concurrent,then $a, b,$ and $c$ are in which of the following?

The equation of the line passing through the point of intersection of the lines $x + 2y + 6 = 0$ and $2x - y = 2$ and making an intercept $5$ on the $y$-axis is

If the lines $4x + 3y - k = 0$,$2x + y + 3 = 0$,and $3x + 2y + k = 0$ are concurrent,then the perpendicular distance from the point of concurrency of these lines to the line $3x + 4y + 2 = 0$ is

The lines $2x + y - 1 = 0$,$ax + 3y - 3 = 0$,and $3x + 2y - 2 = 0$ are concurrent for

If $P \equiv \left( \frac{1}{x_p}, p \right), Q = \left( \frac{1}{x_q}, q \right), R = \left( \frac{1}{x_r}, r \right)$ where $x_k \neq 0$ denotes the $k^{th}$ term of an $H.P.$ for $k \in N$,then:

If the point of intersection of the lines $2ax + 4ay + c = 0$ and $7bx + 3by - d = 0$ lies in the $4^{th}$ quadrant and is equidistant from the two axes, where $a, b, c,$ and $d$ are non-zero numbers, then $ad : bc$ is equal to