Question: Prove that a similarity transformation (replacing $x$ by $tx$ and $y$ by $ty$) carries an ellipse with center at the origin into another ellipse with the same eccentricity.
(The next questions go on to prove the same result (and converses) for hyperbolas.)
Please don't feel the need to read through all my attempts, I just wanted to show what I had tried and give some background on what has been covered in the book.
I think my question boils down to this: am I allowed to say "given some conic, align the coordinate axes so that the origin is at the center of the conic and the x-axis is aligned with the main axes of the conic" without any further justification?
My Attempt
This is easy to prove for an ellipse (or hyperbola) in standard Cartesian form:
$$\frac{x^2}{a^2}+\frac{y^2}{a^2(1-e^2)}=1$$
Replacing $x$ and $y$ by $tx$ and $ty$ respectively, we have
$$\frac{(tx)^2}{a^2}+\frac{(ty)^2}{a^2(1-e^2)}=\frac{x^2}{\left(\frac a t\right)^2}+\frac{(y)^2}{\left(\frac a t\right)^2(1-e^2)}=1$$
Which which is another ellipse with a center at the origin with the same eccentricity. The problem is that the standard form only describes ellipses and hyperbolas with vertical directrices. Intuitively I understand that you could choose the coordinate axes so that any ellipse is described by this equation, but this has not been made explicit in the book.
I suppose one question worth asking, at this point, is if this is all Apostol wanted me to prove? Taking the original question literally, it seems my argument for choosing the coordinate axes is a bit of hand-waving to get at the other cases of ellipses.
EDIT: After discussing this with a professor at the university I attended, his interpretation was that Apostol's (high-level) intention was just to show that scaling the shape of an ellipse does not change the eccentricity, because eccentricity is a description of the shape and not the size. He thought, after reading the chapter, that I should consider only those curves in the ideal standard form for all questions such as this, and if necessary remark (although it is yet unproven in the book) that any conic is congruent to one in standard form. In this case, my original solution is adequate.
EDIT #2: It now seems clear that Apostol's intention was to consider only the standard cases, as the next question is Use the Cartesian equation which represents all conics of eccentricity $e$ and center at the origin to prove that these conics are integral curves of the differential equation $y'=(e^2-1)x/y$. Indeed, it seems that the integral curves of that differential equation are given by $\frac{x^2}{C}+\frac{y^2}{C(1-e^2)}=1$, which are really only the central conics at the origin with horizontal directrices, not all conics at the origin. Since, in that question, he does not make a distinction, I think he is just being atypically loose with his terminology.