Ellipse Formula Derivation

An ellipse usually looks like a squashed circle:

images/geom-ellipse.js?mode=foci

"F" is a focus, "G" is a focus,
and together they are called foci.
(pronounced "fo-sigh")

Ellipse with foci F and G and a point P on the curve.

The total distance from F to P to G stays the same

In other words, we always travel the same distance when going from:

point "F" to
to any point on the ellipse
and then on to point "G"

Definition

An ellipse is the set of all points on a plane whose distance from two fixed points F and G add up to a constant.

Foci on a Graph

An ellipse is defined by its two foci (focus points).

Let's put them on a graph!

Let's place the two foci on the x-axis at:
F = (−c, 0) and G = (c, 0)

For any point P(x, y) on the ellipse, the definition says:

Distance(F, P) + Distance(G, P) = 2a

Using the distance formula on P(x,y), we can write this as:

√(x + c)² + y² + √(x −&minus c)² + y² = 2a

Let's solve!

Start with:

√(x + c)² + y² + √(x − c)² + y² = 2a

Move one radical to right side:

√(x + c)² + y² = 2a − √(x − c)² + y²

Square both sides:

(x + c)² + y² = 4a² − 4a√(x − c)² + y² + (x − c)² + y²

Expand the brackets:

x² + 2cx + c² + y² = 4a² − 4a√(x − c)² + y² + x² − 2cx + c² + y²

Cancel matching terms from both sides:

2cx = 4a² − 4a√(x − c)² + y² − 2cx

Move the −2cx left:

4cx = 4a² − 4a√(x − c)² + y²

Divide everything by 4:

cx = a² − a√(x − c)² + y²

Isolate the radical on the left:

a√(x − c)² + y² = a² − cx

Square both sides again:

a²[(x − c)² + y²] = (a² − cx)²

Expand everything:

a²(x² − 2cx + c² + y²) = a⁴ − 2a²cx + c²x²

Distribute the a²:

a²x² − 2a²cx + a²c² + a²y² = a⁴ − 2a²cx + c²x²

Cancel −2a²cx from both sides:

a²x² + a²c² + a²y² = a⁴ + c²x²

x terms on left, constants on right:

a²x² − c²x² + a²y² = a⁴ − a²c²

Factor out x² and a²:

(a² − c²)x² + a²y² = a²(a² − c²)

Almost there! Now let's define a new variable b where:

a² = b² + c²

b² = a² − c²

Substituting b² into our equation gives:

b²x² + a²y² = a²b²

Beautiful and simple!

To make it more useful, divide every term by a²b² to get:

x²a² + y²b² = 1

The standard ellipse equation

Try the sliders here:

../algebra/images/equation-graph.js?fn0=x^2/a^2 + y^2/b^2 = 1;xmin=-2.5;xmax=2.5;ymin=-1.5;ymax=1.5;vara=2|0|5;varb=1|0|5

a is the semi-major axis (the distance from the exact center (0,0) to the far tip on the x-axis).

b is the semi-minor axis (the distance from the center (0,0) to the very top point on the y-axis).

Hyperbola

Hyperbola on a Cartesian plane with horizontal transverse axis

The formula for the equation of the hyperbola is very similar:

x²a² − y²b² = 1

The standard hyperbola equation

Except for "−" instead of "+".

We can follow a similar route to derive it's formula, too.