Ellipse.pdf
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An ellipse is a bounded plane curve where the sum of the distances from two focal points is a constant. It has the following key parts: - Two focal points located inside the ellipse along its major axis - A center point at the intersection of its major and minor axes - Vertices that are the endpoints of its major axis - Covertices that are the endpoints of its minor axis The standard form of an ellipse's equation is (x-h)2/a2 + (y-k)2/b2 = 1, where a and b are the lengths of the semi-major and semi-minor axes.
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- 2. What is Ellipse? ➢ Bounded curve which looks like a flattened circle Applications: ● Orbits of the planets ● Moons and comets ● Space and Astronomy ● Some airplane wings
- 3. Activity:
- 4. What is Ellipse? ➢ Set of all points in a plane such that the sum of the distances from two points (foci) is a constant. F1(-c,0) and F2(c,0) are Foci ➢ The set of all point P(x,y) whose distance from F1(-c,0) and from F2(c,0) add up to a certain constant is called Ellipse.
- 5. CENTER ❑ The center has coordinate (h,k) or (0,0) ❑ The intersection between the major and minor axis 01 FOCI 02 ❑ Two fixed points inside the ellipse which are equidistance from the center and lays along the major axis. ❑ For any point P in the ellipse, the sum of its distance from the foci is equal to the length of the major axis ELLIPSE has the following parts
- 6. VERTICES ❑ Points on the ellipse, collinear with the center and foci ❑ Endpoint of the major axis and is a units away from the center. Thus the segment from one vertex to the other is 2a 03 COVERTICES 04 ❑ The endpoints of the minor axis, perpendicular to the major axis ❑ The length of the segment from the covertices is 2b ELLIPSE has the following parts
- 7. LATUS RECTUM ❑ The chord that passes through a focus and is perpendicular to the major axis. 05 ELLIPSE has the following parts
- 11. Example
- 12. Example
- 13. Example
- 15. 1. You TRY! Find the standard equation of the following 2. 3. Foci at ( ) and whose any point on it, the sum of its distances is 16
- 17. Example
- 18. Example
- 19. Example
- 20. Find the standard equation of the following 1. Center at (-5,1), a focus at (-2,1) and a vertex at (-1,1) 2. Foci at (0,-7) and (0,3) and vertices at (0,5) and (0,-9) 3. Foci at (-2,0) and (4,0) and whose any point on it, the sum of its distances are 8 You TRY!
- 21. GENERAL FORM of the Equation 𝐴𝑥2 + 𝐵𝑦2 + 𝐶𝑥 + 𝐷𝑦 = F
- 22. Let’s TRY! Write the standard equation of 𝑥2 + 𝑦2 − 8𝑥 − 6𝑦 = −21 Write the standard equation of 2𝑥2 + 9𝑦2 + 16𝑥 − 90𝑦 = −239 Write the standard equation of 2𝑥2 + 9𝑦2 + 16𝑥 − 90𝑦 = −239