An ellipse is a curve where the sum of the distances from two fixed focal points is a constant. It is defined as the set of all points whose distances from two focal points add up to a constant. The standard equation of an ellipse is (x/a)^2 + (y/b)^2 = 1, where the focal points are located at (±c,0) and the vertices are located at (±a,0) and covertices at (0,±b). Examples are given of finding the focal points, vertices, and covertices of ellipses with given standard equations and of writing the standard equation of an ellipse given its focal points and constant sum of distances.

1. ELLIPSE
INTRODUCTION
PRECALCULUS
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2. Introduction
An ellipse is one of the conic sections; Its shape
is a bounded curve which looks like a flattened
circle. The orbits of the planets in our solar
system around the sun happen to be elliptical
in shape. Also, just like parabolas, ellipses have
reflective properties that have been used in the
construction of certain structures.

3. Definition and Equation of an Ellipse
Consider the points F1(3,0) and F2(3,0), as shown in
Figure 1.19. What is the sum of the distances of
A(4,2.4) from F1 and from F2? How about the sum of
the distances of B and C(0,4) from F1 and from F2?

4. REMEMBER THIS!
An ellipse is the set of all points on the plane,
the sum of whose distances from two fixed
points is a constant.

5. Definition and Equation of an Ellipse
Let F1 and F2 be two
distinct points. The set of all
points P, whose distances
from F1 and from F2 add up
to a certain constant, is
called an ellipse. The
pointsF1 and F2 are called
the foci of the ellipse.

6. GRAPH OF AN ELLIPSE WITH STANDARD
EQUATION
Here the features of the graph of an ellipse
with standard equation
where a > b. Let 𝑐 = 𝑎2 − 𝑏2 .
𝑥2
𝑎2
+
𝑦2
𝑏2
= 1

7. GRAPH OF AN ELLIPSE WITH STANDARD
EQUATION
(1) Center: origin (0,0)
(2) foci: F1(-c,0) and F2(c,0)
(3) vertices:
V1(-a,0) and V2(a,0)
(4) covertices
W1(0,-b) and W2(0,b)

8. ELLIPSE WITH STANDARD EQUATION
Example 1. Give the coordinates of the foci,
vertices, and covertices of the ellipse with
equation
𝑥2
25
+
𝑦2
9
= 1.
Sketch the graph, and include these points.

9. ELLIPSE WITH STANDARD EQUATION
Example 2. Find the (standard) equation of
the ellipse whose foci are F1(3,0) and F2(3,0),
such that for any point on it, the sum of its
distances from the foci is 10.

10. TRY THIS!
1. Give the coordinates of the foci, vertices,
and covertices of the ellipse with equation
𝑥2
169
+
𝑦2
25
= 1. Sketch the graph, and include
these points.
2. Find the equation in standard form of the
ellipse whose foci are F1(8,0) and F2(8,0),
such that for any point on it, the sum of its
distances from the foci is 20.

11. THANK YOU
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