College Algebra 9th txtbk

SECTION 6–2 Ellipse 395
6-2 Ellipse
Z Definition of an Ellipse
Z Drawing an Ellipse
Z Standard Equations of Ellipses and Their Graphs
Z Applications
We start our discussion of the ellipse with a coordinate-free definition. Using this definition,
we show how an ellipse can be drawn and we derive standard equations for ellipses
specially located in a rectangular coordinate system.
Z Definition of an Ellipse
The following is a coordinate-free definition of an ellipse:
Z DEFINITION 1 Ellipse
An ellipse is the set of all points P in a plane
d 1 d 2 Constant
such that the sum of the distances from P to
B
two fixed points in the plane is a constant (the
V
d
constant is required to be greater than the
1 P
distance between the two fixed points). Each of F
d
the fixed points, F and F, is called a focus,
2
and together they are called foci. Referring to
F V
the figure, the line segment VV through the
B
foci is the major axis. The perpendicular
bisector BB of the major axis is the minor axis. Each end of the major axis, V
and V, is called a vertex. The midpoint of the line segment FF is called the center
of the ellipse.
Z Drawing an Ellipse
An ellipse is easy to draw. All you need is a piece of string, two thumbtacks, and a pencil
or pen (see Figure 1 on the next page). Place the two thumbtacks in a piece of cardboard.
These form the foci of the ellipse. Take a piece of string longer than the distance between
the two thumbtacks—this represents the constant in the definition—and tie each end to a
thumbtack. Finally, catch the tip of a pencil under the string and move it while keeping the
string taut. The resulting figure is by definition an ellipse. Ellipses of different shapes result,
depending on the placement of thumbtacks and the length of the string joining them.