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lesson4.-ellipse f.pptx
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- 6. What do younotice about the pictures? IdentifyandDescribethePicture
- 7. LEARNINGOBJECTIVES 01 define anellipse 03 graph an ellipse in a rectangular coordinate system given an equation with center at (0,0) 02 determine the standard form of equation of an ellipse with center at (0,0)
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- 10. Definition of Ellipse Anellipse is a set of points such that the sum of the distances from two fixed points is constant. The fixed points are the foci, and the constant sum is the length of major axis. ELLIPSE F F P(x,y) 𝑑1 𝑑2 𝑑1 + 𝑑2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
- 11. The ellipse intersects the majoraxis called vertices, denoted as V1 and V2. The length of the segment V1V2 is equal 2a. PropertiesofEllipse 𝑉1 𝑉2 2𝑎 𝑎
- 12. The midpoint of thesegment V1V2 is called the center of the ellipse, denoted by C. PropertiesofEllipse 𝑉1 𝑉2 𝐶
- 13. The line segment throughthe center perpendicular to the major axis is the minor axis, denoted as B1B2 which are the covertices. The length of the line segment is equal to 2b. PropertiesofEllipse 𝑉1 𝑉2 𝐶 𝐵1 𝐵2 2𝑏 𝑏
- 14. The length ofthe line segment F1F2 is 2c. The line segment through F1 and F2 are called the latera recta and each has a length of 2𝑏2 𝑎 . PropertiesofEllipse 𝑉1 𝑉2 𝐶 𝐵1 𝐵2 𝐹1 𝐹2 2𝑐 𝑐
- 15. The length ofthe line segment F1F2 is 2c. The line segment through F1 and F2 are called the latera recta and each has a length of 2𝑏2 𝑎 . PropertiesofEllipse 𝑉1 𝑉2 𝐶 𝐵1 𝐵2 𝐹1 𝐹2 2𝑏2 𝑎 𝑏2 𝑎 𝐸1 𝐸2 𝐸3 𝐸4
- 16. In ellipse, a>b a>c If a=b,then the ellipse is circle. 𝑐2 = 𝑎2 − 𝑏2 PropertiesofEllipse 𝑉1 𝑉2 𝐶 𝐵1 𝐵2 𝐹1 𝐹2 𝐸1 𝐸2 𝐸3 𝐸4
- 17. Definition of Ellipse Anellipse is a set of points such that the sum of the distances from two fixed points is constant. The fixed points are the foci, and the constant sum is the length of major axis. ELLIPSE F F P(x,y) 𝑑1 𝑑2 𝑑1 + 𝑑2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2𝑎
- 18. GENERALFORMANDFORMULAS 𝑨𝒙𝟐 + 𝑪𝒚𝟐+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 GeneralformofanEllipseequation whereinBismissing,andthe numericalcoefficientsof𝑥2 and𝑦2 havethesamesigns.
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- 21. EXAMPLE1 Sketch the graphand determine the coordinates of the foci, vertices, and covertices 𝒙𝟐 𝟐0 + 𝒚𝟐 36 = 𝟏
- 22. EXAMPLE 2 Sketch thegraph and determine the coordinates of the foci, vertices, and covertices 𝟐𝟓𝒙𝟐 + 𝟒𝒚𝟐 = 𝟏𝟎𝟎
- 23. EXAMPLE 3 Find andgraph the equation of the ellipse a which satisfies the following conditions: Foci at (±2,0); one vertex at (4,0)
- 24. EXAMPLE4 Sketch the graphand determine the coordinates of the foci, vertices, and covertices 𝒙𝟐 9 + 𝒚𝟐 16 = 𝟏
- 25. ACTIVITY5 A. Sketch thegraph and determine the coordinates of the foci, vertices, and covertices of this ellipse. 𝑥2 16 + 𝑦2 25 = 1 B. Determine the standard form of equation of an ellipse given: Vertices (0, ±5); minor axis of length 6
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- 27. 𝒙 − 𝒉𝟐 𝒂𝟐 + 𝒚 − 𝒌 𝟐 𝒃𝟐 = 𝟏 𝒙 − 𝒉 𝟐 𝒃𝟐 + 𝒚 − 𝒌 𝟐 𝒂𝟐 = 𝟏 VERTEXAT(h,k)
- 28. EXAMPLE1 Give the coordinatesof the foci, vertices, and covertices (𝒙 + 3)𝟐 𝟐4 + (𝒚 − 5)𝟐 4𝟗 = 𝟏
- 29. EXAMPLE1 Give the coordinatesof the foci, vertices, and covertices of the given general form of ellipse. 𝟗𝒙𝟐 + 𝟏𝟔𝒚𝟐 − 𝟏𝟐𝟔𝒙 + 𝟔𝟒𝒚 = 𝟕𝟏
- 30. EXAMPLE1 The foci ofan ellipse are (−3,−6) and (−3, 2). For any point on the ellipse, the sum of its distances from the foci is 14. Find the standard equation of the ellipse and give the coordinates of the center, vertices and covertices.
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Editor's Notes
- #21 Always remember, the major axis contains the center, the foci, and the vertices.
- #22 Balik sa definition. The sum of the focal distances of any point on the ellipse is constant and that is equal to 2a. (PF1 + PF2 = 2a)
- #30 Optional. Pwedeng isang at origin or isang hk
- #31 Optional. Pwedeng isang at origin or isang hk