Ellipses were introduced in
Introduction to Conic Sections,
as one of
several different curves (‘conic sections’) that are formed by intersecting a plane with an
infinite double cone.
Identifying Conics by the
Discriminant
introduced the general equation for any conic
section,
and gave conditions under which the graph would be an ellipse.
In this current section, we present and explore the standard definition of an ellipse.
This definition facilitates the derivation of standard equations for ellipses.
Recall that the notation ‘$\,d(P,Q)\,$’ denotes the distance between points $\,P\,$ and $\,Q\,.$
DEFINITION
ellipse
An ellipse is the set of points in a plane such that
the sum of the distances to
two fixed points
is constant. More precisely:

$\,P\,$ is a general point on the ellipse. $\,d(P,F_1) + d(P,F_2) = \text{constant}$ 
Got a piece of cardboard, paper, tape, string/cord (not stretchy), and pen/pencil? Then, you can create your own ellipse:
I grew my own sunflowers from seed in 2017, when I was writing this section!) 
You can also play with ellipses using the dynamic JSXGraph at right:


In the definition of ellipse, the ellipse constant $\,k\,$ is
required to be
strictly greater than the distance between the two foci.
Why?
As shown below, other values of $\,k\,$ don't give anything that
a reasonable person would want to call an ellipse!
A ‘LINE SEGMENT’ ELLIPSE: $\,k = d(F_1,F_2)\,$Suppose the ellipse constant, $\,k\,,$ equals the distance between the foci:that is, $\,k = d(F_1,F_2)\,.$ In this case, the solution set to the equation $$ \cssId{s62}{\color{green}{d(P,F_1)} + \color{red}{d(P,F_2)} = k} $$ is the line segment between $\,F_1\,$ and $\,F_2\,$ (including the endpoints). Most people don't want to call a line segment an ellipse! This is why $\,k\,$ is not allowed to equal $\,d(F_1,F_2)\,$ in the definition of ellipse. 

AN ‘EMPTY’ ELLIPSE: $\,k < d(F_1,F_2)\,$The shortest distance between any two points is a straight line.In particular, the shortest distance from $\,F_1\,$ to $\,F_2\,$ is the length of the line segment between them, and is denoted by $\,d(F_1,F_2)\,.$ Thus, any path from $\,F_1\,$ to $\,F_2\,$ must have length greater than or equal to $\,d(F_1,F_2)\,.$ In particular (refer to sketch at right), the piecewiselinear path from $\,F_1\,$ to $\,P\,$ and then from $\,P\,$ to $\,F_2\,$ always has length greater than or equal to $\,d(F_1,F_2)\,.$ Therefore, if the ellipse constant $\,k\,$ is strictly less than $\,d(F_1,F_2)\,,$ there are no points $\,P\,$ that make the following equation true: $$ \cssId{s79}{\overbrace{d(P,F_1) + d(P,F_2)\strut }^{\text{always $\,\,\ge\,\, d(F_1,F_2)$}}} \qquad \cssId{s80}{=}\qquad \cssId{s81}{\overbrace{\strut k}^{< \,\, d(F_1,F_2)}} $$ You might want to call this an empty ellipse, an invisible ellipse, or an imaginary ellipse! There's nothing there! 

On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
IN PROGRESS 