Explanation

(b)
Equation belonging to both families will pass through two fixed points. First intersection point of lies $x + 2y = 0$ and
$3x + 2y + 1 = 0\ is\ \left( - \frac{1}{2},\frac{1}{4} \right)$ and second interception point of lines $x - 2y = 0$ and $x - y + 1 = 0$, is $( - 2,\ - 1)$
Line passing through $\left( - \frac{1}{2},\frac{1}{4} \right)\$and $( - 2,\ - 1)$ is
$y - \frac{1}{4} = \frac{- 1 - \frac{1}{4}}{- 2 + \frac{1}{2}}\left( x + \frac{1}{2} \right)$
$\Rightarrow \ 5x - 6y + 4 = 0$