solving geometric problems analyticially by fitting a coordinate system to them

Question

I encountered a few problems where it is quite helpful to fit the coordinate system to the problem, and I wanted to check here if that's a sound thing to do.

For example, in Tom Apostol's Calculus ex. $13.25.15$ we should prove that the collection of all parabolas is invariant under a similarity transform.

Coming up with a general parabola equation is hard (link), and I would much rather approach the problem like this:

1. Take any parabola.
2. Take the coordinate system such that the axis of the parabola is the X axis, and the directrix is parallel to the Y axis.
3. In such a coordinate system we know the equation of the parabola to be $\tag{1} (y-y_0)^2 = 4c_0(x-x_0)$
4. In $1$, we replace $x$ with $tx$ and $y$ with $ty$ to see that we still have a parabola $(y-y_1)^2 = 4c_1(x-x_1)$.

Is that a correct analytic solution? Is there a more analytical solution? Is there some caution I should be aware of when fitting in coordinate system to similar problems?

Thanks!