Explanation

(c)
The equation of a plane through the line of intersection of the planes $ax + by + c\mathcal{z} + d = 0$
and $a^{'}x + b^{'}y + c^{'}\mathcal{z} + d^{'} = 0$ is
$\left( ax + by + c\mathcal{z} + d \right) + \lambda\left( a^{'}x + b^{'}y + c^{'}\mathcal{z} + d^{'} \right) = 0$
or $x\left( a + \lambda a^{'} \right) + y\left( b + \lambda b^{'} \right) + \mathcal{z}\left( c + \lambda c^{'} \right) + d + \lambda d^{'} = 0$ (i)
This is parallel to $x$-axis, i.e., $y = 0,z = 0$. Therefore,
$1\left( a + \lambda a^{'} \right) + 0\left( b + \lambda b^{'} \right) + 0\left( c + \lambda c^{'} \right) = 0$
$\Rightarrow \lambda = - \frac{a}{a^{'}}$
Putting the value of $\lambda$ in (i), the required plane is $y\left( a^{'}b - ab^{'} \right) + \mathcal{z}\left( ac^{'} - a^{'}c \right) + a^{'}d - ad^{'} = 0$ or $\left( ab^{'} - a^{'}b \right)y + \left( ac^{'} - a^{'}c \right)\mathcal{z} + ad^{'} - a^{'}d = 0$