# Ellipse - Interactive Graphs

You can explore various ellipse graphs on this page, and see the effect of changing parameters (by dragging various points around).

For background information on what's going on, and more explanation, see:

## a. Interactive Graph - Ellipse as a Locus

We learned on The Ellipse page that an ellipse is the locus of (or the "path traced out by") a point where the sum of the distances from 2 fixed points is a constant.

You can explore what this means in the following JSXGraph (it's not a fixed image).

In this case the equation of the ellipse is:

`x^2/64+y^2/25=1`

An ellipse has 2 focus points, shown as points F_{1} and F_{2} (these points are fixed for this first interactive).

### Things to do

You can drag point P around the ellipse.

You can use this to investigate the property that Length PF_{1} + Length PF_{2} is constant for a particular ellipse.

In this example, PF_{1} + PF_{2} = 16.

## b. Interactive Graph - Ellipse with Center other than the Origin

In this next graph, you can vary the **center** of the ellipse to better understand how this changes the **equation** of the ellipse.

We're using the same ellipse as the above example, but changing the center.

At the start, the center of the ellipse is at (8, 2), so the equation of the ellipse is:

`((x-8)^2)/64+((y-2)^2)/25=1`

### Things to Do

**Drag point C**, the center of the ellipse, to see how changing the center of the ellipse changes the equation.

Center: (8, 2) Equation: `(x-p)^2/64 + (y-q)^2/25 = 1`

## c. Eccentricity

In this next graph, you can vary the **eccentricity** of the ellipse by changing the position of the focus points, or of one of the points on the ellipse.

Before exploring the next one, recall:

- Eccentricity = `c/a` is a measure of how elongated the ellipse is. This number ranges from value 1 (where the ellipse is very elongated) to 0 (where the ellipse is actually a circle).
*a*is the distance from the center of the ellipse to the furthest vertex (either of the 2 far vertices).*b*is the distance from the center of the ellipse to the closest vertex (either of the 2 close vertices).*c*is the distance from the center of the ellipse to the focus (either focus).

### Things to do

**Drag** point named 'F_{1}', (one of the focus points for our ellipse) left or right to change the shape (and therefore the eccentricity) of the ellipse.

**Drag** point P (a point on the ellipse) up or down to change the shape (and therefore the eccentricity) of the ellipse.

What shape does the ellipse become when you place the 2 focus points at the origin?

Eccentricity `=c/a = ` Equation of ellipse: `x^2/m + y^2/n = 1`

Go back to « The Ellipse.