Conic section of parabola

Question

The linear eccentricity is the distance between the center and the focus. Why parabola's conic section does not have linear eccentricity?

Answer 1

If we have a constant $A > 0$ and parabola $$ y = A x^2, $$ we can make a geometrically "similar" figure with new coordinates $$ y = \frac{v}{A}, \; \; x = \frac{u}{A}, $$ $$ \frac{v}{A} = A \left( \frac{u}{A} \right)^2 $$ $$ \frac{v}{A} = \frac{u^2}{A}, $$ $$ v = u^2. $$ That is, all parabolas are geometrically similar, in the same sense as we discuss similar triangles.

The same does not hold for ellipses and hyperbolas, which have distinguished centers, and which are not all similar; same thing, not all triangles are similar.