Why is the eccentricity of a parabola 1?

Question

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 "$a$"?

Answer 1

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.