Solving an Ellipse
Below is a picture that relates the foci, eccentricity, and length of the major and minor axes.
We can derive the identities if we recall that if P is a point on the ellipse the sum of the distances from P to the two foci is a constant. We will derive the equation for the ellipse with vertical major axis. The derivative for the horizontal major axis case is similar.
First consider the top of the ellipse. The sum of the distances from the two foci to the top of the ellipse is
(b - c) + (b + c) = 2b
The square of this sum is given by
Next consider the right side of the ellipse. Notice that the distances from each focus to the right side is the same. The square of this sum is
(2d)2 = 4d2
by the Pythagorean theorem,
d2 = a2 + c2
Substituting, and setting these equal to each other gives
4b2 = 4(a2 + c2)
b2 = a2 + c2
To use the applet below follow these steps
When you have mastered the above tutorial, please answer the following in a few complete sentences