Chapter 4: The Ellipse
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1 Chapter 4: The Ellipse SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
2 Chapter 4: The Ellipse Lecture 1: Introduction to Ellipse Lecture 13: Converting General Form to Standard Form of Ellipse and Vice- Versa Lecture 14: Graphing Ellipse with Center at the Origin C (0, 0) Lecture 15: Graphing Ellipse with Center at C (h, k) Lecture 16 : The Ellipse and the Tangent Line
3 Lecture 1: Introduction to the Ellipse SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
4 Explain the Definition of Ellipse Using the Figure Below:
5 Definition of the Ellipse An ellipse is a set of all points P (x, y) in a plane, the sum of whose distances from two specified fixed points F 1 and F is a constant.
6 Definition of the Foci These are the two fixed points (F 1 and F ) of the ellipse.
7 Symbols for Foci F 1, F, to denote the foci of the ellipse.
8 Definition of an Ellipse The ellipse can also be defined as the locus of points whose distance from the focus is proportional to the horizontal distance from a directrix, where the ratio is 0 is less than x but x is less than 1.
9 Two Types of Ellipse Horizontal Ellipse Vertical Ellipse
10 The Horizontal Ellipse When the foci are on the x-axis or parallel to the x- axis, then the ellipse is horizontal.
11 The Vertical Ellipse When the foci are on the y-axis or parallel to the y- axis, then the ellipse is vertical.
12 Classroom Task 11: Using the definition of ellipse and parts of it, derive the standard equation of horizontal ellipse with vertex at the origin.
13 Standard Equation of the Horizontal Ellipse with Vertex at the Origin: The Standard Equation of the Horizontal Ellipse with Vertex at the Origin: x b y 1 a
14 The Horizontal Ellipse with Vertex at the Origin:
15 Representations: Let: P, be the point in the plane/ ellipse with P(x, y); x = x and y = y; F 1, be one of the fixed points with F 1 (c, 0); x = c and y = 0; F, be one of the fixed points with F (-c, 0); x = -c and y = 0; PF 1, be the distance from P(x, y) and F1(c, 0); PF, be the distance from P(x, y) and F(-c, 0); and k, be the sum of the distances of PF 1 and PF which is constant.
16 The Horizontal Ellipse with Vertex at the Origin:
17 Something to think about What is the relationship of the sum of the two sides of a triangle to its third side?
18 The Relationship: In a triangle, the sum of the lengths of the two sides is GREATER THAN the third side.
19 Standard Equation of the Horizontal Ellipse with Vertex at the Origin: The Standard Equation of the Horizontal Ellipse with Vertex at the Origin: x b y 1 a
20 The Horizontal Ellipse with Vertex at the Origin:
21 Standard Equation of the Vertical Ellipse with Vertex at the Origin: The Standard Equation of the Vertical Ellipse with Vertex at the Origin: y b x 1 a
22 Classroom Task 10: Family Activity 1: Ellipse in the Real World SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
23 Instructions: Please prepare a five-minute presentation about the applications of ellipse in the real-world by family. The presentation can be in a form of a family report, skit, video presentation, etc. This will be presented on our next meeting.
24 Grading System: Criteria Percentage Content 40 Organization of Ideas 0 Communication Skills 15 Presentation and Aesthetic Consideration Behavior during the Presentation 15 10
25 TED Ed Video (Nice to Know): Lithotripsy: Ellipse in the Medicine Field
26 Civil Engineering: Elliptical Bridge
27 Aeronautical Engineering: British Spitfire
28 Automobile: Elliptical Gears
29 Medicine: Medical Lithotripsy
30 Naval Architecture: Racing Sailboat
31 Astronomy: Planetary Motion
32 The Ellipse: Parts of the Graph of the Ellipse SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
33 Definition of the Axis of Symmetry The line that passes through both foci is the axis of symmetry and meets at two points called vertices.
34 Symbols for Vertices V 1,V, to denote the vertices of the ellipse.
35 Definition of the Major Axis The line segment joining the vertices and the foci is called the major axis. It is also called as traverse axis and has a length of a.
36 The Semi-Major Axis The letter a is called the semi-major axis of the ellipse.
37 Definition of the Minor Axis The line segment which is a perpendicular bisector of the major axis is called minor axis. It is also called as conjugate axis with a length of b.
38 Definition of the Minor Axis The line segment which is a perpendicular bisector of the major axis is called minor axis. It is also called as conjugate axis with a length of b.
39 The Semi-Minor Axis The semi-minor axis is the value of b in the length of minor axis of an ellipse which is b.
40 Definition of the Center The center of an ellipse is the intersection of the major axis and the minor axis.
41 Definition of the Center C, to denote the center of the ellipse.
42 Definition of the Directrix It is a line such that the ratio of distance of the points on the conic section from focus to its distance from the directrix is constant.
43 Symbols for Directrices D 1, D, to denote the directrices of the ellipse.
44 Definition of the Latera Recta The plural form of latus rectum, is the chord that passes through the focus, and is perpendicular to the major axis and has both endpoints on the curve.
45 Symbols for the Endpoints of Latera Recta E 1, E, E 3, E 4 to denote the endpoints of the latera recta of the ellipse.
46 Other Symbols: a, is the distance from the center to the vertex; b, is the distance from the center to one endpoint of the minor axis;
47 Other Symbols: c, is the distance from the center to the focus; e, is the value of the eccentricity
48 Other Symbols: a, is the length of the major axis; and b, is the length of the minor axis.
49 The Ellipse: Properties of the Ellipse SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
50 Property Number 1 of Ellipse: The length of the major axis is a.
51 Property Number of Ellipse: The length of the minor axis is b.
52 Property Number 3 of Ellipse: The length of the latus rectum is b a.
53 Property Number 4 of Ellipse: The center is the intersection of the axes.
54 Property Number 5 of Ellipse: The endpoints of the major axis are called the vertices.
55 Property Number 6 of Ellipse: The endpoints of the minor axis are called the co-vertices.
56 Property Number 7 of Ellipse: The line segment joining the vertices is called the major axis.
57 Property Number 8 of Ellipse: The line segment joining the co-vertices is called the minor axis.
58 Property Number 9 of Ellipse: The eccentricity of the ellipse is 0 e 1.
59 Theorem 4.: The Eccentricity (e) of the Ellipse The eccentricity (e) of an ellipse is the ratio of the undirected distance between the foci to the undirected distance between vertices; that is: e c a.
60 Classroom Task 1: We say that circle is a special type of an ellipse. Can you prove that circle is a special type of an ellipse?
61 What do we know about their radii?
62 Performance Task 1: Please download, print and answer the Let s Practice 1. Kindly work independently.
63 Lecture 1: Converting General Form of the Ellipse to Its Standard Form and Vice-Versa SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
64 What should you expect? This section represents how to convert general form of ellipse to its standard form and vice-versa.
65 Table 4.1: Equations of Ellipse Center Major Axis (0, 0) x-axis (0, 0) y-axis (h, k) (h, k) x-axis y-axis Cx Cx Ax Ax General Form A C Cy F 0, A C Ay F 0 A C Cy Dx Ey F 0, Ay Dx Ey F 0 A C y a Standard Form x a ( x h) a a b ( y k) a y 1, a b b x 1 a b b ( y k) b ( x h) b 1, 1 a b
66 Example 3: Convert the following general equations to standard form: 9x 8y 88
67 Final Answer The standard form is: x y
68 Example 33: Convert the following general equations to standard form: 5x 49 y 1,5
69 Final Answer The standard form is: x y
70 Example 34: Convert the following general equations to standard form: 3x 4 y 4x 16 y 5 0
71 Final Answer The standard form is: ( x 4) ( y 4 ) 3 1
72 Conclusion 1 about the Ellipse: When the radius of the ellipse is of positive sign, then the ellipse exists.
73 Example 35: Convert the following general equations to standard form: 9x 6 y 54 x 4 y 110 0
74 Final Answer The standard form is: 9( x 3) 6( y ) 5
75 Something to think about What can you observe on the right side of the equation? What can you conclude?
76 Conclusion about the Ellipse: When the radius of the ellipse is of negative sign, then the ellipse does not exist.
77 Example 36: Convert the following general equations to standard form: 9x 4 y 7 x 4 y 180 0
78 Final Answer The standard form is: 9( x 4) 4( y 3) 0
79 Something to think about What can you observe on the right side of the equation? What can you conclude?
80 Conclusion 3 about the Ellipse: When the radius of the ellipse is of zero value, then the ellipse will degenerate to a point.
81 Example 37: Convert the following standard form to general form: ( y ) ( x 5 3) 9 1
82 Final Answer The general form is: 5 x 9 y 150 x 36 y 36 0
83 Example 38: Convert the following standard form to general form: ( x 1) ( y 1)
84 Final Answer The general form is: 36 x 100 y 7 x 00 y 3,464 0
85 Performance Task 13: Please download, print and answer the Let s Practice 13. Kindly work independently.
86 Lecture 13: Graphing an Ellipse with Center at the Origin SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
87 Learning Expectation: This section presents how to graph an ellipse and how to determine the parts of an ellipse where the center is at the origin.
88 Table 4. : Parts of the Graph of Ellipse with Center at the Origin Standard Equation Foci Vertices Covertices Endpoints of Latera Recta Directrix F 1 (c, 0) F (-c, 0) V 1 (a, 0) V (-a, 0) B 1 (0, b) B (0, -b) F 1 (0, c) F (0, -c) V 1 (0, a) V (0, -a) B 1 (b, 0) B (-b, 0) a b c E a b c E a b c E a b c E 4 3 1,,,, c a x c a b E c a b E c a b E c a b E,,,, c a y 1 b y a x 1 b x a y
89 Example 39: Sketch and discuss the following equation of an ellipse: 5 x 9 y 5
90
91 Example 40: Find the equation of the ellipse with center at C (0, 0), length of major axis is 10 units, and a focus at F 1 (4, 0). Identify the parts of the ellipse and sketch its graph.
92 Final Answer: The equation of the ellipse is: x y
93
94 Example 41: Find the equation of the ellipse with center at C (0, 0), vertices at (4,0), and eccentricity e 3. Identify the parts of the ellipse and sketch its graph.
95 Final Answer: The equation of the ellipse is: x y
96
97 Performance Task 14: Please download, print and answer the Let s Practice 14. Kindly work independently.
98 Lecture 14: Graphing an Ellipse with Center at C (h, k) SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
99 Learning Expectation: This section presents how to graph an ellipse and how to determine the parts of an ellipse.
100 Example 4: Sketch and discuss the following equation of an ellipse: 36 x 100 y 7 x 00 y 3,464 0
101
102 Example 43: Find the equation of the ellipse with center at C (-4, 7), a focus at F 1 (-4, 11) and a vertex at V (-4, 1). Identify the parts of the ellipse and sketch its graph.
103 Final Answer: The equation of the ellipse is: or5 x ( y 7) ( x 4) 5 9 9y 00 x 16 1 y
104
105 Example 44: Find the equation of the ellipse with center at C (, -3), vertices at V 1 (7, -3) and V (-3, -3), and eccentricity of e = 3/5. Identify the parts of the ellipse and sketch its graph.
106 Final Answer: The equation of the ellipse is: or16 x ( x ) ( y 3) y 1 64 x 150 y
107
108 Performance Task 15: Please download, print and answer the Let s Practice 15. Kindly work independently.
109 Lecture 15: Tangent to the Ellipse SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza
110 Tangential to the Ellipse A tangent to the ellipse is a line that touches the ellipse at just one point.
111 Tangent to the Ellipse
112 The Equation of the Line The equation of the line can be determined using the formula: y mx a m b
113 Example 45: Find the equation of the tangent to the ellipse and the line x 4y 36 passes at a point PT (1, 3). Sketch its graph.
114 Final Answer: The equation of the tangent line is: x 3y 15 0.
115
116 Example 46: Find the point on the ellipse x 5y 36 which is the closest, and which is the farthest point from the line x 5y 30 Sketch the graph. 0.
117 Final Answer: The closest point is (-4, ) and the farthest point is (4, -) to the line x 3y 15 0.
118
119 Performance Task 15: Please download, print and answer the Let s Practice 15. Kindly work independently.
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