Relation Between a, b, and c

Where c is the focal distance (distance from centre to focus) and e is the eccentricity. Both forms are equivalent; the second is used when switching between geometric and eccentricity-based descriptions.

From the derivation of the standard form, $b^2 = a^2 - c^2$, equivalently:

$$c^2 = a^2 - b^2$$

Geometric meaning: Consider the right triangle formed by the centre $O$, the co-vertex $(0, b)$, and the focus $(c, 0)$. The distance from the co-vertex to the focus is:

$$\sqrt{c^2 + b^2} = \sqrt{(a^2-b^2)+b^2} = a$$

So the distance from either co-vertex to either focus is exactly $a$ — the semi-major axis. This provides a quick geometric construction: to locate the foci, draw arcs of radius $a$ from the co-vertices; they intersect the major axis at the foci.

In terms of eccentricity: $c = ae$, so:

$$b^2 = a^2 - c^2 = a^2(1-e^2)$$

This form is useful when $e$ is given instead of $c$.