Elements of the ellipse

Once the equation is written in canonical form, every element of the ellipse can be read off directly from the two denominators $a^2$ and $b^2$.

Property — Elements of the ellipse

From the canonical equation $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ with $a > b > 0$:

• Semi-axes: $a$ (major, along $x$), $b$ (minor, along $y$).
• Foci: $F_1(-c;0)$, $F_2(c;0)$ with $c = \sqrt{a^2 - b^2}$.
• Vertices: $(\pm a; 0)$ and $(0; \pm b)$.
• Eccentricity: $e = c/a$, with $0 \le e < 1$. The closer $e$ is to $0$, the “rounder” the ellipse (tending to the circle); the closer $e$ is to $1$, the more “flattened” the ellipse.
• Directrices: $x = \pm\tfrac{a}{e} = \pm\tfrac{a^2}{c}$, vertical lines external to the ellipse.

Beware of the case in which $a < b$ (semi-major axis along $y$): all the roles are swapped. The foci lie on the $y$-axis, with $c = \sqrt{b^2 - a^2}$, the eccentricity becomes $e = c/b$, and so on. It is always best to compare $a^2$ and $b^2$ before saying where the foci lie.

Links

Topics: Ellipse
Concepts: Directrices · Eccentricity · Ellipse · Foci · Semi-axes · Vertices
Skills: Analytic geometry · Using formulae