I was not able to relate why we do not get a change in eccentricity when we change the leading coefficient in the equation of a parabola, which in most cases gives us a measure of relative "steepness" or "deformation" of a curve. The more we increase the absolute value of the leading coefficient, the steeper and more "deformed" our parabola becomes (we are basically making its shape more and more deformed as compared to a circle, whose eccentricity is 0). Can we relate this to the change in the parabola's eccentricity?
2026-05-05 18:05:31.1778004331
As we change the leading coefficient in the equation of a parabola, y=ax^2+bx+c are we changing its eccentricity? If not, then why?
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The eccentricity does not change. It is always $e=1$ for a parabola. As you increase $a$, both the focus and the directrix get closer to the vertex, but the eccentricity remains $=1$.