1. Ellipses
On to Sec. 8.2a…

2. Geometry of an Ellipse
An ellipse is the set of all points in a plane whose
distances from two fixed points in the plane have a
constant sum. The fixed points are the foci (plural
of focus) of the ellipse. The line through the foci is
the focal axis. The point on the focal axis midway
between the foci is the center. The points where the
ellipse intersect its axis are the vertices of the ellipse.

3. Focal Axis
Focus
Focus
Vertex
Center
Vertex
The sum
is a constant

4. Deriving the Equation of an Ellipse
Something to notice:
Equation for the two
dashed distances:
Use the distance formula:

5. Deriving the Equation of an Ellipse

6. Deriving the Equation of an Ellipse
This equation is the standard form of the equation of an ellipse
centered at the origin with the x-axis as its focal axis.
When the y-axis is the focal axis?
Chord – segment with endpoints on the ellipse
Major Axis – chord lying on the focal axis
Minor Axis – chord through the center, perp. to the focal axis
The number a is the semimajor axis, and the number b is the
semiminor axis.

7. Ellipses with Center (0, 0)
Standard Equation:
(0, b)
Focal Axis: x-axis a b
Foci: c
(–a, 0)
(–c, 0)
(c, 0)
(a, 0)
Vertices:
Semimajor Axis: a
Semiminor Axis: b
(0, –b)
Pythagorean Relation:

8. Ellipses with Center (0, 0)
Standard Equation:
(0, a)
Focal Axis: y-axis
(0, c) a
Foci: c
Vertices:
Semimajor Axis: a
(–b, 0) b
(b, 0)
Semiminor Axis: b
(0, –c)
Pythagorean Relation:
(0, –a)

9. Some Practice Problems…
Find the vertices and the foci of the ellipse
First, write in standard form:
Because the larger number is under the “x” term, the focal axis
is the x-axis. So…
Foci:
Vertices:

10. Some Practice Problems…
Find an equation of the ellipse with foci (0, –3) and (0, 3) whose
minor axis has length 4. Sketch the ellipse and support your
sketch with a grapher.
The center is (0,0)
The foci are on the y-axis with c = 3
The semiminor axis is b = 4/2 = 2
Use the Pythagorean relationship to solve for a:

11. Some Practice Problems…
Find an equation of the ellipse with foci (0, –3) and (0, 3) whose
minor axis has length 4. Sketch the ellipse and support your
sketch with a grapher.
Standard Form:
Can we graph by hand?
To check with a calculator, solve for y:

12. What happens when the center of an ellipse is not on the origin?
(h – c, k)
(h + c, k)
(h, k)
(h, k + a)
(h, k + c)
(h + a, k)
(h – a, k)
(h, k)
(h, k – c)
(h, k – a)

13. Ellipses with Center (h, k)
Standard Equation
Focal Axis
Semimajor Axis
Foci
Semiminor Axis
Vertices
Pythagorean Relation

14. Ellipses with Center (h, k)
Standard Equation
Focal Axis
Semimajor Axis
Foci
Semiminor Axis
Vertices
Pythagorean Relation

15. Guided Practice
Find the standard form of the equation for the ellipse whose
major axis has endpoints (–2, –1) and (8, –1), and whose minor
axis has length 8.
How about starting with a diagram of the given info?
What is the general equation?
Where is the center  midpoint of the major axis!!!

16. Guided Practice
Find the standard form of the equation for the ellipse whose
major axis has endpoints (–2, –1) and (8, –1), and whose minor
axis has length 8.
Now, find the semimajor and semiminor axes:
Plug all of these values back into our general equation!!!

17. Whiteboard Practice (–2, 5) (–2, 12), (–2, –2)
Find the center, vertices, and foci of the ellipse
Center:
(–2, 5)
Standard form:
Vertices:
Vertices:
(–2, 12), (–2, –2)
Foci:
(–2, ), (–2, –1.325)
Foci:

18. Whiteboard Practice Standard Form:
Find an equation in standard form for the ellipse that satisfies the
given conditions.
Major axis endpoints are (–2, –3) and (–2, 7), minor axis length 4
Start with a diagram…
Center:
Semimajor and semiminor axes:
Standard Form:

19. Whiteboard Practice Standard Form:
Find an equation in standard form for the ellipse that satisfies the
given conditions.
The foci are (–2, 1) and (–2, 5); the major axis endpoints are
(–2, –1) and (–2, 7)
Start with a diagram…
Center:
Semimajor axis:
Semiminor axis:
Standard Form: