Can a line be a taxicab parabola?

Question

For example, a line segment can be a taxicab ellipse if the sum of the distances equals the distance between the foci.

So, can a line be a taxicab parabola?

Answer 1

If the focus is on the directrix, then the "parabola" is the perpendicular to the directrix through the focus.

Answer 2

If we define the "taxicab" directions as parallel to the $x$ and $y$ axes, we can let the $x$ axis be the didectrix and (0,2) be the focus. A line from $(0,1)$ to $(2,2)$, then $y=x$ for all $x>2$, will be half of a taxicab parabola, the other half being the reflection through the $y$ axis.