1. Conic Sections: Ellipses
Pre–Calculus PreAP/Dual, Revised ©2015
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4.6 - Fundamental Theorem of Algebra

2. Shape of an ellipse from a cone
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11.1: Ellipses

3. Real-Life Examples
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11.1: Ellipses

4. 4.6 - Fundamental Theorem of Algebra
Definitions
Ellipse: Set of points whose sum of the distances from two fixed points is consistent
Foci: Segment point joining the vertices given at a point. It is always with the MAJOR axis
Major Axis: The longer line segment of the two segments
Minor Axis: The smaller line segment of the two segments
Vertices: Endpoints of the major axis
Co-Vertices: Endpoints of the minor axis
Latus Rectum: A line segment through the foci of the shape in which it is perpendicular through the major axis and endpoints of the ellipse
Eccentricity: Ratio to describe the shape of the conic 0 < e < 1
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4.6 - Fundamental Theorem of Algebra

5. Formulas to know:
Horizontal Axis Standard Form: If the major axis is on the x-axis (horizontal)
If the major axis is on the x-axis (horizontal)
a is associated with major axis.
Foci of the major axis:
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11.1: Ellipses

6. Formulas to know:
Vertical Axis Standard Form: If the major axis is on the y-axis (vertical)
If the major axis is on the y-axis (vertical)
a is associated with major axis.
Foci of the major axis:
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11.1: Ellipses

7. All Standard Form Equations
Formulas to know:
All Standard Form Equations
Center
Length of Major Axis
Length of Minor Axis
Foci Equation
Length of Latus Rectum
Eccentricity
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11.1: Ellipses

8. Brief Clip
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11.1: Ellipses

9. Horizontal Ellipse (0, b) (–a, 0) (a, 0) (h, k) (0, –b)
(–c, 0) F
(c, 0) F
(–a, 0)
(a, 0)
(h, k)
(0, –b)
Center: (h, k) Foci: (+c, 0) Length of Major Axis: 2a Vertices: (+a, 0) Length of Minor Axis: 2b Co-Vertices: (0, +b) Length of Latus Rectum: 2b2/a Latus Rectum: (+c, +b2/a) Eccentricity: c/a
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11.1: Ellipses

10. Vertical Ellipse (0, a) (b, 0) (–b, 0) (h, k)F
(c, 0)
(b, 0)
(–b, 0)
(h, k)
(–c, 0) F
Center: (h, k) Foci: (0, +c) Length of Major Axis: 2a Vertices: (0, +a) Length of Minor Axis: 2b Co-Vertices: (+b, 0) Length of Latus Rectum: 2b2/a Latus Rectum: (+b2/a, +c) Eccentricity: c/a
(0, –a)
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11.1: Ellipses

11. When in trouble…
GRAPH and PLOT
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11.1: Ellipses

12. 4.6 - Fundamental Theorem of Algebra
Example 1
Graph 𝒙 𝟐 𝟐𝟓 + 𝒚 𝟐 𝟒 =𝟏 A: B: C:
Type:
Center:
Vertices:
Co-Vertices:
Foci:
Latus Rectum:
Length of Major Axis:
Length of Minor Axis:
Length of Latus Rectum:
Eccentricity: F F
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4.6 - Fundamental Theorem of Algebra

13. 4.6 - Fundamental Theorem of Algebra
Example 2
Graph 𝟏𝟔 𝒙 𝟐 +𝟒 𝒚 𝟐 =𝟔𝟒 A: B: C:
Type:
Center:
Vertices:
Co-Vertices:
Foci:
Latus Rectum:
Length of Major Axis:
Length of Minor Axis:
Length of Latus Rectum:
Eccentricity: F F
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4.6 - Fundamental Theorem of Algebra

14. 4.6 - Fundamental Theorem of Algebra
Your Turn
Graph 𝒙 𝟐 𝟒𝟗 + 𝒚 𝟐 𝟖𝟏 =𝟏 A: B: C:
Type:
Center:
Vertices:
Co-Vertices:
Foci:
Latus Rectum:
Length of Major Axis:
Length of Minor Axis:
Length of Latus Rectum:
Eccentricity: F F
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4.6 - Fundamental Theorem of Algebra

15. Steps of Writing Ellipses Equation
Identify the values of A, B, and C.
Plot/draw the figure with the given information if possible
Write the equation and label the needed information
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4.6 - Fundamental Theorem of Algebra

16. Example 3
Write an equation in standard form for an ellipse with center (0, 0), vertex at (6, 0) and co-vertex at (0, 4).
Identify the values of a and b.
The vertex (6, 0) gives the value of a.
a = 6
The co-vertex (0, 4) gives the value of b.
b = 4
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11.1: Ellipses

17. Example 3
Write an equation in standard form for an ellipse with center (0, 0), vertex at (6, 0) and co-vertex at (0, 4).
a = 6
b = 4 x2 a2
= 1 y2 b2 62 42
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11.1: Ellipses

18. Example 4
Write an equation in standard form for an ellipse with center (0, 0), vertices at (+5, 0) and co-vertices at (0, +3).
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11.1: Ellipses

19. Your Turn
Write an equation in standard form for an ellipse with center (0, 0), vertex at (0, 9) and co-vertex at (5, 0).
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11.1: Ellipses

20. Example 5
An ellipse has its center located in the origin. Find the equation where the foci points are at (+8, 0), length of minor axis is 4 units long and graph.
A =
B =
C = ?? 2 8 F F
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11.1: Ellipses

21. Example 5
An ellipse has its center located in the origin. Find the equation where the foci points are at (+8, 0), length of minor axis is 4 units long and graph.
A =
B =
C = ?? 2 8
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11.1: Ellipses

22. Example 5
An ellipse has its center located in the origin. Find the equation where the foci points are at (+8, 0), length of minor axis is 4 units long and graph.
A =
B =
C =
2√17 2 8 F F
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11.1: Ellipses

23. Example 5
An ellipse has its center located in the origin. Find the equation where the foci points are at (+8, 0), length of minor axis is 4 units long and graph.
A =
B =
C =
2√17 2 8
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11.1: Ellipses

24. Example 5
An ellipse has its center located in the origin. Find the equation where the foci points are at (+8, 0), length of minor axis is 4 units long and graph.
A =
B =
C =
2√17 2 8
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11.1: Ellipses

25. Your Turn
An ellipse has its center at the origin. Find an equation of the ellipse with the ends of the major axis points at (0, +5), and the foci of (0, +4).
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11.1: Ellipses

26. Example 6 Graph (𝒙+𝟔) 𝟐 𝟗 + (𝒚−𝟓) 𝟐 𝟒 =𝟏 F F Type: Center: Vertices:B: C:
Type:
Center:
Vertices:
Co-Vertices:
Foci:
Latus Rectum:
Length of Major Axis:
Length of Minor Axis:
Length of Latus Rectum:
Eccentricity: F F
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11.1: Ellipses

27. Your Turn Graph (𝒙−𝟐) 𝟐 𝟒 + (𝒚+𝟏) 𝟐 𝟏 =𝟏 F F Type: Center: Vertices:B: C:
Type:
Center:
Vertices:
Co-Vertices:
Foci:
Latus Rectum:
Length of Major Axis:
Length of Minor Axis:
Length of Latus Rectum:
Eccentricity: F F
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11.1: Ellipses

28. 4.6 - Fundamental Theorem of Algebra
Example 7
An ellipse with the center of (–3, 5) has vertices at (–3, 10) and (–3, 0) and its foci is at (–3, 8) and (–3, 2). Determine the equation. F F
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4.6 - Fundamental Theorem of Algebra

29. 4.6 - Fundamental Theorem of Algebra
Example 8
A conic has foci points of (0, 0) and (4, 0). The major axis length is 8. Determine the equation.
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4.6 - Fundamental Theorem of Algebra

30. 4.6 - Fundamental Theorem of Algebra
Example 9
A conic has the vertices at (5, 0) and (5, 12). The endpoints of the minor axis are at (1, 6) and (9, 6). Determine the equation.
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4.6 - Fundamental Theorem of Algebra

31. 4.6 - Fundamental Theorem of Algebra
Your Turn
A conic has the co-vertices at (2, 3) and (2, 1). The length of the major axis is 4. Determine the equation.
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4.6 - Fundamental Theorem of Algebra

32. 4.6 - Fundamental Theorem of Algebra
Standard Form
Identify whether it is an ellipse by using the equation, b2 – 4ac where the answer is less than zero using the equation, Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
Rearrange variables for x’s and y’s through factoring
Take GCF and add everything to other side
Use completing the square; using what’s added to the x’s and y’s is added to the radius
Identify the coefficient which is raised to the first power and divide the term by 2
Take the second term, divide the term by 2, and square that number
Add to both sides to the equation
Put the equation into factored form
Put it in standard form
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4.6 - Fundamental Theorem of Algebra

33. 4.6 - Fundamental Theorem of Algebra
Example 10
Change the equation x2 + 4y2 + 10x – 8y + 13 = 0 to standard form.
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4.6 - Fundamental Theorem of Algebra

34. 4.6 - Fundamental Theorem of Algebra
Example 10
Change the equation x2 + 4y2 + 10x – 8y + 13 = 0 to standard form.
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4.6 - Fundamental Theorem of Algebra

35. 4.6 - Fundamental Theorem of Algebra
Example 10
Change the equation x2 + 4y2 + 10x – 8y + 13 = 0 to standard form.
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4.6 - Fundamental Theorem of Algebra

36. 4.6 - Fundamental Theorem of Algebra
Example 10
Change the equation x2 + 4y2 + 10x – 8y + 13 = 0 to standard form.
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4.6 - Fundamental Theorem of Algebra

37. 4.6 - Fundamental Theorem of Algebra
Example 11
Change the equation 4x² + 9y² – 32x + 36y + 64 = 0 to standard form.
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4.6 - Fundamental Theorem of Algebra

38. 4.6 - Fundamental Theorem of Algebra
Your Turn
Change the equation 4x² + 9y² – 32x + 36y + 64 = 0 to standard form.
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4.6 - Fundamental Theorem of Algebra

39. 4.6 - Fundamental Theorem of Algebra
Assignment
Worksheet
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4.6 - Fundamental Theorem of Algebra