Lesson-9Intro-of-Ellipse.pptx An ellipse is a special type of curve formed by all points in a plane where the sum of the distances from two fixed points, called foci, is constant

An ellipse is a special type of curve formed by all points in a plane where the sum of the distances from two fixed points, called foci, is constant

1. PRE calculus AMADO R. GUMANGAN TEACHER III

2. Opening Prayer Dear God, we give thanks to you, our Lord and Savior Jesus Christ, for another beautiful day with our loved ones. As we conduct our day, you are forever in our hearts and thoughts. We ask for your love, guidance, and protection in everything we do. Please watch over those who mean the most to us. We ask all these things, in Jesus Name, AMEN.

3. content / lesson DEFINITION OF AN ELLIPSE

4. introduction on ellipse An ellipse is one of the conic sections that most students have not encountered formally before, unlike circles and parabolas. Its shape is a bounded curve which looks like a flattened circle. The orbits of the planets in our solar system around the sun happen to be elliptical in shape.

5. introduction on ellipse Also, just like parabolas, ellipses have reflective properties that have been used in the construction of certain structures (shown in some of the practice problems). We will see some properties and applications of ellipses in this section.

6. Illustrations of Ellipse in Real - Life

7. ACTIVITY   Present an activity. Show me!   Based on the given example above , show or identify more illustrations of an ellipse in real life- situations.

8. POSSIBLE ANSWERS  

9. ACTIVITY   Ellipses abound in nature. The orbits of the planets around the sun is an ellipse. Comets also follow an elliptical path. When a tilted plane cuts a right circular cone and the curve is bounded, or the curve has no opening, such a curve is an ellipse.(see Figure 1).

10. DEFINITION . The Ay writing tstart the lesson by writing the word “TANGENT LINE” on the bond asking the class what they know about a tangent line or where they first heard the word “ “TANGENT LINE” on the board and asking the class what they know about a tangent line or where they first heard the word “tangent”. Ellipses abound in nature. The orbits of the planets around the sun is an ellipse. Comets also follow an elliptical path. When a tilted plane cuts a right circular cone and the curve is bounded, or the curve has no opening, such a curve is an ellipse. (see Figure 1).

11. DEFINITION OF an eLLIPSE . The Ay writing tstart the lesson by writing the word “TANGENT LINE” on the bond asking the class what they know about a tangent line or where they first heard the word “ “TANGENT LINE” on the board and asking the class what they know about a tangent line or where they first heard the word “tangent”. An ellipse is a set of all points 𝑃 (𝑥, 𝑦) on a plane so that the sum of their distances from two distinct points 𝐹1 and 𝐹2 (the foci) is a constant.

12. Discuss the properties and elements of an Ellipse. Remarks: Aside from the set of points and foci mentioned in the definition, we also have the following: 1. Center 𝐶 (ℎ, 𝑘). 2. Vertices, 𝑉1 and 𝑉2. The segment, 𝑉1𝑉2 , is called the major axis of the ellipse. 3. |𝑎| is the distance from the center to a vertex (The length of the major axis is 2𝑎).

13. Discuss the properties and elements of an Ellipse. 4. The segment perpendicular to the major axis passing through the center whose endpoints are on the ellipse is the minor axis. 5. |𝑏| is the distance from the center to an endpoint of the minor axis (The length of the minor axis is 2𝑏).

14. Discuss the properties and elements of an Ellipse. 6. 𝑎 > b > 0 7. |𝑐| is the distance from the center to a focus. 𝐹1𝐹2 equals 2c. 8. 𝑎 > 𝑐 > 0

15. Equation of an Ellipse Horizontal axis 1. Since 𝑎 is the distance from the center to a vertex, then the vertices have coordinates 𝑉 (ℎ ± 𝑎, 𝑘). 2.Also, since c is the distance from the center to a focus, then the foci has coordinates 𝐹 (ℎ ± 𝑐, 𝑘).

16. Equation of an Ellipse Vertical axis 1. Since 𝑎 is the distance from the center to a vertex, then the vertices have coordinates 𝑉 (ℎ, 𝑘 a). 2.Also, since c is the distance from the center to a focus, then the foci has coordinates 𝐹 (ℎ , 𝑘 ± 𝑐 ).  

17. PROPERTIES OF AN ELLIPSE

18. Some practical application of concepts and skills in daily living real-life. Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves.

19. Directions: Read the given questions carefully and answer each item correctly. ___1. What are the coordinates of a covertex if the vertex has a coordinate (ℎ − 𝑎, 𝑘)? A. (ℎ + 𝑏, 𝑘) B. (ℎ − 𝑐, 𝑘) C. (ℎ, 𝑘 + 𝑏) D. 𝑉 (ℎ, 𝑘 + 𝑐) _____2. What are the coordinates of a focus if a covertex has coordinates (ℎ, 𝑘 − 𝑏)? A. (ℎ, 𝑘 + 𝑎) B. (ℎ + 𝑎, 𝑘) C. (ℎ, 𝑘 + 𝑐) D. (ℎ + 𝑐, 𝑘)

20. ___ 3. What are the coordinates of the covertex if a vertex is located to the right of the center? A. (ℎ, 𝑘 + 𝑏) B. (ℎ, 𝑘 − 𝑐) C. (ℎ + 𝑏, 𝑘) D. (ℎ + 𝑐, 𝑘) _____4. What are the coordinates of the covertex if the focus is found below the vertex? A. 𝑉 (ℎ, 𝑘 + 𝑏) B. 𝑉 (ℎ, 𝑘 + 𝑐) C. 𝑉 (ℎ + 𝑏, 𝑘) D. 𝑉 (ℎ + 𝑐, 𝑘) _____5. What are the coordinates of the covertex if the minor axis is vertical? A. (ℎ, 𝑘 + 𝑏) B. (ℎ + 𝑏, 𝑘) C. (ℎ + 𝑎, 𝑘) D. (ℎ, 𝑘 + 𝑎)

21. Thank you Xer madz