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What I know:

The standard form for a parabola is $(x-h)^2=4a(y-k)$, if the axis of symmetry of the parabola is vertical, or $(y-k)^2=4a(x-h)$, if the axis of symmetry of the parabola is horizontal.

The "$a$" value is the distance from the vertex $(h, k)$ to the focus.

So for the eccentricity to be $1$, $c/a = 1$, therefore "$c$" must be equivalent to "$a$".

My question:

How do you represent "$c$" in a parabola? Is "$c$" the same line as "$a$"?

caverac
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BOB COM
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    To answer the title, you may want to resort to polar representation of conic sections. The parabolas follows if $e=1$ is chosen. – imranfat May 26 '18 at 20:55
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    This answer might be useful. – robjohn May 26 '18 at 21:14
  • See also this answer for a discussion in the context of cones. There, we see that the eccentricity of a parabola is $1$ because the "cutting plane" makes the same angle as the cone. – Blue May 26 '18 at 22:49
  • The $a$ in your parabola formula has a different meaning from the $a$ in the ellipse/hyperbola context (where it measures major vertex distance to the center). The center of a parabola is a point at infinity (on the axis of symmetry), therefore the analogs to $a,c$ from the elliptic/hyperbolic case would be infinite in the parabolic case. – ccorn May 26 '18 at 22:55
  • @robjohn so it's like infinity/infinity = 1 ? – theenigma017 Dec 25 '18 at 19:19

1 Answers1

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A well known property of conic sections (ellipse, parabola or hyperbola) is as follows:

A conic section is the locus of points whose distance from a given point (focus) is proportional to the distance from a given line (directrix). The fixed proportionality ratio $\epsilon$ is the eccentricity.

For $\epsilon<1$ the locus defined above is an ellipse, for $\epsilon=1$ a parabola and for $\epsilon>1$ a hyperbola.

Intelligenti pauca
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