# Conic section of parabola

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

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.