Lecture #7 analytic geometry
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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.
Lecture #7 analytic geometry
- 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
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- 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