We have seen how the equation of an ellipse can be derived using the distance formula. Now, let's explore how the parameters (h, k, a, and b) change the shape of our ellipse. Play with the applet below to try and figure out how they affect the shape.

Notice that the ellipse equation looks similar to the circle equation where x^{2} + y^{2} = 1 changes to (x/a)^{2} + (y/b)^{2} = 1 and that is the standard equation for an ellipse centered at the origin.

The center is the starting point at (h,k).

The major axis contains the foci and the vertices.

Major axis length = 2a. This is also the constant that the sum of the distances must add to be. (In the case above, 2|P| or 2Q works as well, since |P| = Q

Minor axis length = 2b. (in the case above, we have 2R or 2|S| and R = |S|)

Distance between foci = 2c. (once again, in our case, 2F)

Notice, the foci are within the curve.

Since the vertices are the farthest away from the center, a is the largest of the three lengths, and the Pythagorean relationship is: a^{2} = b^{2} + c^{2}.

Eccentricity

Eccentricity: the measure of how circular an ellipse is (or how "squashed" it is). 0 E 1such that,
E = 0 --> circle
E = 1 --> flat line
Eccentricity =
Move around the sliders to see how the eccentricity changes. When does the eccentricity = 0? When does it equal 1?