Ellipses

By the end of this lesson, you will be able to write the standard form equation of an ellipse, identify center, vertices, co-vertices, and foci, and apply ellipses to orbital mechanics and other contexts.

The Geometric Definition

An ellipse is the set of all points for which the sum of distances from two fixed points (the foci) is constant.

$$PF_1 + PF_2 = 2a = \text{constant}$$

The sum equals the length of the major axis ($2a$).

Standard Forms

Horizontal major axis ($a^2$ under $x^2$, $a > b$):

$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$

• Center: $(h, k)$
• Vertices (major): $(h \pm a, k)$
• Co-vertices (minor): $(h, k \pm b)$
• Foci: $(h \pm c, k)$ where $c^2 = a^2 - b^2$

Vertical major axis ($a^2$ under $y^2$, $a > b$):

$$\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$$

• Vertices: $(h, k \pm a)$
• Co-vertices: $(h \pm b, k)$
• Foci: $(h, k \pm c)$

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

$a$ = semi-major axis length (distance from center to vertex)

$b$ = semi-minor axis length (distance from center to co-vertex)

$c$ = distance from center to focus

The foci are always between the center and the vertices: $c < a$.

Eccentricity: $e = c/a$ (ranges from 0 for a circle to approaching 1 for a very elongated ellipse).

Example 1 — Finding All Features

$\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$

Center: $(0,0)$; $a^2 = 25 \to a = 5$; $b^2 = 9 \to b = 3$

$c = \sqrt{25-9} = \sqrt{16} = 4$

Vertices: $(\pm 5, 0)$ (horizontal major axis since $a$ under $x^2$)

Co-vertices: $(0, \pm 3)$

Foci: $(\pm 4, 0)$

Example 2 — Translated Ellipse

$\dfrac{(x+2)^2}{16} + \dfrac{(y-3)^2}{4} = 1$

Center: $(-2, 3)$; $a^2 = 16 \to a = 4$; $b^2 = 4 \to b = 2$

$c = \sqrt{12} = 2\sqrt{3}$

Horizontal major axis: vertices $(-2 \pm 4, 3) = (2,3)$ and $(-6,3)$

Foci: $(-2 \pm 2\sqrt{3}, 3)$

Example 3 — Writing the Equation

Ellipse with foci $(0, \pm 3)$ and vertices $(0, \pm 5)$.

Vertical major axis (foci on $y$-axis). $c = 3$, $a = 5$, $b = \sqrt{25-9} = 4$.

$$\frac{x^2}{16} + \frac{y^2}{25} = 1$$

Example 4 — From Standard to Equations (Completing the Square)