Ellipse - Standard and General Form, and Graphs

Ellipse

1. Ellipse

2. Objectives: Define an ellipse. Determine the standard form of equation of an ellipse. Graph an ellipse in a rectangular coordinate system Solve situational problems.

3. Elliptical Roads

4. Architectural Designs

5. Logo Designs

6. Medical Equipment LITHOTRIPTER

7. An ellipse is a closed figure in a plane that closely resembles an oval.

8. An image of an ellipse can be formed if a plane cuts a right circular cone and the plane is not parallel to any generator.

9. An ellipse is a set of all coplanar points such that the sum of its distances from two fixed points is constant. The fixed points are called the foci of the ellipse.

10. Parts of an Ellipse: An ellipse has two axes of symmetry. The longer axis is called the major axis , and the shorter axis is called the minor axis . The major axis contain the foci which are inside of the ellipse.

11. Center Co-vertex Co-vertex V ertex V ertex Focus Focus Latus Rectum Latus Rectum Minor Axis Major Axis

12. Center Co-vertex Co-vertex V ertex V ertex Focus Focus Latus Rectum Latus Rectum a a b b c c  

13. a a b b c c        

14. The intersection of the two axes is called the center. The center is the midpoint between the two foci, and also the midpoint between the two vertices. The ellipse intersects the major axis in two points called the vertices of the ellipse. These vertices are the endpoints of the major axis.

15. A segment passing through a focus of the ellipse that is perpendicular to the major axis is called a latus rectum. A line outside the ellipse that is parallel to the minor axis and has the same distance from a vertex as the focus is the directrix. Since there are two foci, there are also two directrices.

16. Standard Equation of Ellipse with center at c(0,0)

17. Ellipse with center at c(0,0) Center: Origin (0,0) 2. Foci: F 1 (-c, 0) and F 2 (c, 0) - Each focus is c units away from center. - For any point on the ellipse, the sum of its distance from the foci is 2a.  

18. For any point on the ellipse, the sum of its distance from the foci is 2a.

19. Ellipse with center at c(0,0) 3. Vertices: V 1 (-a,0) and V 2 (a,0) - The vertices are points on the ellipse, collinear with the center and foci. - If y = 0, then x = +-a. Each vertex is a units away from C. - The segment V 1 V 2 is called the major axis. Its length is 2a.

20. Ellipse with center at c(0,0) 4. Covertices: W 1 (0,-b) and W 2 (0,b) -The segment through the center, perpendicular to the major axis is the minor axis. -If x=0, then y=+-b. Each covertex is b units away from the center. -The minor axis W 1 W 2 is 2b units long. Since a > b, the major axis is longer than the minor axis.

21. Ellipse with center at c(0,0) 5. End of the latus rectum ( , +- and ( , +-  

22. Example: Give the coordinates of the foci, vertices, covertices and directrices of the ellipse with equation. Sketch the graph. Solution: By inspection on the equation, we can say that the Center is at (0,0)with horizontal major axis. Step 1: Find a and b: a=5 Vertices: (-5,0) and (5,0) b=3 Covertices: (0,-3) and (0,3) Step 2: Find c: = = 4  

23. Example: Give the coordinates of the vertices, covertices, foci and directrices of the ellipse with equation. Sketch the graph. Step 2: Find c: = = 4 F: (-4,0) and (4,0) Step 3: Find the directrices: d = a – c = 5 – 4 = 1 Since line VF is equidistant to VD then D: x = -6 , x =6  

24. D: x=-6 D: x=6

25. Example 2: Find the standard equation of the ellipse whose foci are F 1 (-3,0) and F 2 (3,0) such that for any point on it, the sum of the distances from the foci is 10. Answer:  

26. Seatwork: Give the coordinates of the vertices, covertices, foci and directrices of the ellipse with the equation. Sketch the graph. 2. Find the equation in standard form of the ellipse whose foci are F1 (-8,0) and F2 (8,0), such that for any point on it, the sum of the distances from foci is 20.  

27. Assignment: Find the coordinates of the foci and directrices, the endpoints of the major axis, minor axis and the latus rectum for each ellipse whose center (0,0). Draw the ellipse, its foci and directrices.  

28. Quiz: Write the equation of the ellipse with center at the origin that satisfies the given conditions. Draw the ellipse, its foci and directrices. The foci have coordinates (+-4,0) and a vertex at (5,0). b. The length of the latus rectum is and the vertices have coordinates (0,-5) and (0,5). Hint: (+- or LR = 2  

29. Ellipse with Center at ( h,k )

30. Ellipse with Center at ( h,k )

31. Note that if is the denominator of , the major axis is horizontal. If is the denominator of the , the major axis is vertical. The gen form of the equation Where A>0, C>0 and A≠C. However, there are other cases with the same general form that are not ellipse. These are degenerate conics.  

32. 10 a week 300 100 water 125 floorwax 125 floorwax

33. Example: Give the coordinates of the center, foci, vertices and covertices of the ellipse with the given equation. Sketch the graph and include these points. 1. 2.  

34. Solution1. The ellipse is vertical. From = 49 then a=7, b= and c = c =5 Center: (-3, 5) Foci: (-3, 10) and (-3, 0) Vertices: (-3,12) and (-3,-2) Covertices: (-3-2 , 5) or (-7.89 , 5) and (-3+2 , 5) or (1.89,5)  

35. The ellipse is vertical. From = 49 then a=7, b= and c = c =5 Center: (-3, 5) Foci: (-3, 10) and (-3, 0) Vertices: (-3,12) and (-3,-2) Covertices: (-3-2 , 5) or (-7.89 , 5) and (-3+2 , 5) or (1.9,5 )  

36. Solution 2. We first change the given equation to standard form. The ellipse is horizontal. From = 64 then a=8, b= b=6, and c = c = = = 5.3  

37. Center: (7, -2) Foci: (1.71, -2) and (12.29, -2) Vertices: (15,-2) and (-1,-2) Covertices: (7,4) and (7,-8)

38. Example 3: Express each equation in standard form. Determine the center, foci, vertices, covertices and directrices. Find the length of the minor axis, major axis and latus rectum. 4  

39. Situational Problem 1 A tunnel has the shape of semi ellipse that is 15 ft high at the center, and 36 ft across the base. At most how high should a passing truck be, if it is 12 ft wide, for it to be able to fit through the tunnel?

40. Situational Problem 1 A road tunnel with a semi elliptical arch has 16 m wide base a 6 m high altitude at the center. How close to the either wall of the tunnel can a vehicle that is 2 m high pass by the tunnel?

41. Seatwork: The foci of an 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. Give the coordinates of the center, foci, vertices and covertices of the ellipse with equation . Sketch the graph and include these points. An ellipse has vertices of (-10, -4) and (6, -4) and covertices (-2,-9) and (-2, 1). Find its standard equation and its foci.  

42. Assignment: Give the coordinates of the center, vertices, covertices, and foci of the ellipse with the given equation. Write your answers in table form as below. Sketch the graph, and include these points.

43. Quiz: A. Write the equation of the ellipse in standard form that satisfies the given conditions. Draw the ellipse, its foci and directrices. 1. The foci have coordinates (+-4,0) and a vertex at (5,0). 2. The center is at (7, -2), a vertex at (2,-2) and an endpoint of a minor axis at (7, -6). 3. The vertices are at (-2, -2) and (-2, 8) and the length of the minor axis is 6. 4. The center is at (4,3), the length of the horizontal major axis is 5, the length of the minor axis is 4. 5. Foci (-7,6) and (-1,6), the sum of the distances of any point from the foci is 14.

44. Quiz: B. Express each equation in standard form. Determine the center, foci, vertices, covertices, end points of latus rectum and directrices. Find the length of the minor axis, major axis and latus rectum.  

45. Quiz: C. The arch of a bridge is in the shape of semi ellipse, with its major axis at the water level. Suppose the arch is 20 ft high in the middle, and 120 ft across its major axis. How high above the water level is the arch, at a point 20 ft from the center (horizontally)?

46. Thank you and God bless!