I have a general question: when given the parametric form of a line such that:
$$ x = a_1 + \lambda t_1 $$
$$ y = a_2 + \lambda t_2 $$
$$ z = a_3 + \lambda t_3 $$
The Cartesian form is:
$$ \frac{x-a_1}{t_1} = \frac{y-a_2}{t_2} = \frac{z-a_3}{t_3} $$
What happens when one of the $t$ is zero? For example, let $t_2 = 0$. Then what happens?
When you set one of the $t_i$ to be zero you need to look at the original equations first before considering the Cartesian form. In this case, if $t_2 = 0$ then the second equation becomes $y=a_2$ which is still the equation of a line -- but the Cartesian form with this is just $$\frac{x-a_1}{t_1} = \lambda = \frac{z-a_3}{t_3} \quad \mbox{and} \quad y = a_2$$
This technique of considering alternate forms of an equation to help you understand the behaviour at singular points (e.g. where the denominator of a fraction becomes zero) is used moderately often in mathematics.