
Consider an ellipse with foci $\,F_1\,$ and $\,F_2\,$ and ellipse constant $\,k\,$.
Follow along with the sketch at right as you read these observations.

By the definition of ellipse, every point $\,P\,$ on the ellipse
satisfies:
$$ d(P,F_1) + d(P,F_2) = k$$
Here, $\,d(A,B)\,$ denotes the distance between points $\,A\,$ and $\,B\,$.

Vertices of an Ellipse:
Let $\,I_1\,$ and $\,I_2\,$ denote the
intersection points of:
 the ellipse
 the line through the foci
By definition, $\,I_1\,$ and $\,I_2\,$ are called the vertices of the ellipse.

Major Axis of an Ellipse:
By definition,
the major axis of the ellipse is the line segment from $\,I_1\,$ to $\,I_2\,$.
The major axis is shown at right in green.

The distance from $\,I_1\,$ to $\,F_1\,$ is the
same as the distance from $\,I_2\,$ to $\,F_2\,$.
Here's a proof:
Define $\,s_1 := d(I_1,F_1)\,$, $\,s_2 := d(I_2,F_2)\,$, and $\,t := d(F_1,F_2)\,$.
Since $\,I_1\,$ is on the ellipse, $\,d(I_1,F_1) + d(I_1,F_2) = k\,$.
That is, $\,s_1 + (s_1 + t) = k\,$.
Similarly, since $\,I_2\,$ is on the ellipse, $\,s_2 + (s_2 + t) = k\,$.
Solving $\,2s_1 + t = k\,$ and $\,2s_2 + t = k\,$ simultaneously gives $\,s_1 = s_2 := s\,$.

the Length of the Major Axis is the Ellipse Constant:
Using the proof above, the length of the major axis is $\,2s + t = k\,$.
Thus, an ellipse ‘shows’ its ellipse constant as the length of its major axis!

Center of an Ellipse:
Let $\,C\,$ denote the midpoint of the major axis.
By definition, $\,C\,$ is called the center of the ellipse, and is shown at right in red.

Minor Axis of an Ellipse:
Draw the line through the center that is perpendicular to the major axis.
Let $\,I_3\,$ and $\,I_4\,$ denote the intersection points of this line with the ellipse.
By definition, the minor axis of the ellipse is the line segment from $\,I_3\,$ to $\,I_4\,$.
The minor axis is shown at right in purple.

The Minor Axis is shorter than the Major Axis:
Focus attention on the yellow right triangle.
The hypotenuse has length $\,\frac k2\,$. (Be sure that you can explain why!)
Let $\,r := d(I_3,C)\,$.
Note that the length of the minor axis is $\,2r\,$.
Since each leg of a right triangle is strictly shorter than the hypotenuse:
$$\begin{gather}
r < \frac k2\cr\cr
\overbrace{2r}^{\text{length of minor axis}} <
\overbrace{k}^{\text{length of major axis, ellipse constant}}
\end{gather}
$$

$I_1\,$ and $\,I_2\,$ are the vertices of the ellipse;
they are the intersection points of
the ellipse and the line through the foci.
The major axis of the ellipse
is the line segment that connects the vertices.
Its length is the ellipse constant, $\,k\,$.
An ellipse ‘shows’ its ellipse constant
as the length of its major axis!
The center of an ellipse
is the midpoint of the major axis.
You might roughly want to think of the major axis
as giving the ‘length’ of the ellipse,
and the minor axis (the segment from $\,I_3\,$ to $\,I_4\,$)
as giving the ‘width’ of the ellipse.
The major axis is always longer than the minor axis.
