An ellipse is defined by two focal points, such that the sum of the distances from any point on the ellipse to the two foci is a constant. An ellipse has two axes - a major axis and a minor axis. There are different forms for the equation of an ellipse depending on the position of its center and orientation of its axes. Formulas are provided to calculate the coordinates of vertices, foci, and axes lengths for ellipses in standard position or with arbitrary position and orientation.

1. Lecture #7
Ellipse
Parts of Ellipse and its graph
• Equation of Ellipse
- Standard Equation
- General Equation
• Formulas

2. ELLIPSE
 An ellipse is defined by two points, each called a
focus. If you take any point on the ellipse, the sum of
the distances to the focus points is constant.

3. PARTS OF AN ELLIPSE
 Vertices – the points at which an ellipse makes its
sharpest turns and lies on the major axis, also end of
major axis
 Co-vertices – ends of minor axis
 Focus/foci – point/s that define the ellipse and lies on
the major axis
 Major axis – the longest diameter of the ellipse
 Minor axis – the shortest diameter of the ellipse

5. EQUATIONS OF ELLIPSE
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6. EQUATIONS OF ELLIPSE
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7. EQUATIONS OF ELLIPSE
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8. EQUATIONS OF ELLIPSE
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9. 

10. FORMULAS
(if center is at the origin and major axis at x-
axis)
Vertices Co-vertices
(a, 0) (-a, 0) (0, b) (0, -b)
Foci Length of LR
(c, 0) (-c, 0)
Length of major and minor axis
2a (major) 2b (minor)
Ends of Latera recta

11. FORMULAS
(if center is at the origin and major axis at y-
axis)
Vertices Co-vertices
(0, a) (0, -a) (b, 0) (-b, 0)
Foci Length of LR
(0, c) (0, -c)
Length of major and minor axis
2a (major) 2b (minor)
Ends of Latera recta

12. FORMULAS
(if center is at (h, k) and major axis at x-axis)
Vertices Co-vertices
(h + a, k) (h - a, k) (h, k + b) (h, k-b)
Foci Length of LR
(h + c, k) (h - c, k)
Length of major and minor axis
2a (major) 2b (minor)
Ends of Latera recta

13. FORMULAS
(if center is at (h, k) and major axis at y-axis)
Vertices Co-vertices
(h, k + a) (h, k - a) (h + b, k) (h - b, k)
Foci Length of LR
(h, k + c) (h, k - c)
Length of major and minor axis
2a (major) 2b (minor)
Ends of Latera recta

14. Sample Problem
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