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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!

S11n
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    This is pretty easy to make completely analytic. Find a coordinate transformation between the two different planes – tryst with freedom Jul 15 '23 at 12:30
  • @SBrian Thanks! I suppose the two different planes are $P_1, P_2$ with coordinate system axis $P_1={(1, 0), (0, 1)}$ and $P_2={(a,b), (c,d)}$, where $(a,b)$ is the parabola axis wrt $P_1$, and $(c,d)$ a vector parallel to the directrix wrt $P_1$. However, I still don't understand why do we even need that, and what prevents us from starting right away with that coordinate system $P_2$, with parabola vertex at $(0,0)$ wrt $P_2$? If possible, it would be nice if you can provide a full answer, so we can also close the question. – S11n Jul 17 '23 at 07:50

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