SHS Pre-Calculus Quarter 1: Week 3 - Module 4 Pre-Calculus Grade 11 Quarter 1: Week 3-Module 4 First Edition, 2020 Copyright © 2020 La Union Schools Division Region I All rights reserved. No part of this module may be reproduced in any form without written permission from the copyright owners. Development Team of the Module Author: Antoniette G. Padua, MT I Editor: SDO La Union, Learning Resource Quality Assurance Team Illustrator: Ernesto F. Ramos Jr., P II Management Team: ATTY. Donato D. Balderas, Jr. Schools Division Superintendent Vivian Luz S. Pagatpatan, Ph.D Assistant Schools Division Superintendent German E. Flora, Ph.D, CID Chief Virgilio C. Boado, Ph.D, EPS in Charge of LRMS Erlinda M. Dela Peña, Ed.D, EPS in Charge of Mathematics Michael Jason D. Morales, PDO II Claire P. Toluyen, Librarian II Pre-Calculus Quarter 1: Week 3-Module 4 TARGET Intersections of a double right circular cone and a plane can make the conic sections and the degenerate cases. There are four conic sections namely the Circle, Parabola, Ellipse and the Hyperbola. Apart from the conic sections are three degenerate cases which includes the Point, the Line and the Intersecting Lines. In the previous lessons, you have learned about the definition of the Circle and the Parabola. You were also taught of the standard equations of the Circle and the Parabola. In this module, you will be provided with information and activities that will help you learn about Ellipses (Definition and Equation). After going through this module, you are expected to: 1. define an ellipse (STEM_PC11AG-Ic-1); and 2. determine the standard form of equation of an ellipse (STEM_PC11AG-Ic-2) Before moving on, check how much you know about this topic. Answer the pre-test provided below: Direction: Write the letter of the correct answer on the blank provided before the number. ______ 1. What is the definition of an ellipse? A. A set of points whose distance from a fixed point called the center is a certain constant. B. A set of points whose distances from two fixed points called the foci add up to a certain constant. C. A set of points whose distances from two fixed points called the foci differs to a certain constant. D. A set of points whose distances from a fixed point called the focus and a fixed line called the directrix is a certain constant. ______ 2. Which among the following cases shows a horizontal ellipse? A. One focus at (-1,-1) center at (-1,1) B. Vertices located at (±5, 0), foci at (±3,0) C. Vertices located at (0, ±5), foci at (0, ±3) D. Covertices located at (±4,0), foci at (0, ±3) 1 ______ 3. What is the standard equation of a horizontal ellipse with center at the origin? A. C. B. D. ______ 4. What is the standard equation of a vertical ellipse with center at the origin? A. C. B. D. ______ 5. What is the standard equation of a horizontal ellipse with center at the point (h, k)? A. t t t C. B. t D. ______ 6. What is the standard equation of a vertical ellipse with center at the point (h, k)? C. t t t C. D. t D. ______ 7. What is the standard equation of an ellipse with foci at (±4,0); length of major axis 10? A. B. t t t t D. t t t C. t t t ______ 8. What is the standard equation of an ellipse with center at (0,0), focus at (0,3) and vertex at (0,5)? A. B. t + t + A. t t t t =1 C. =1 D. t t t t + + =1 t =1 ______ 9. What is the standard equation of an ellipse with center at (-1,0), a vertex at (-4,0) and a focus at (-3,0)? B. t + + t t =1 C. =1 D. t - t - t =1 t =1 - t ______ 10. What is the standard equation of an ellipse with center at (-2,2), a vertex at (-2,4), a focus at (-2,3)? A. B. t t + + t t =1 C. =1 D. 2 t t - t =1 =1 Module 4 ELLIPSE Figure 1: Orbit of the Planets Revolving Around the Sun Ellipse is a topic taken on Mathematics subjects. It is specifically being studied on Analytic Geometry and Calculus. But this concept is also essential in the study of Space and Astronomy. Figure 1 shows Kepler’s Law of Planetary Motion. The law says that every planet’s orbit is an ellipse with the Sun as one of the foci. In module 1, you have learned about the Conic Section. In this module, you will learn about one of the conic sections – ellipse. Like what you have learned with the conic sections circle and parabola, you will also define an ellipse and learn about its standard equation. JUMPSTART Let’s begin! To be able to understand the lesson well, do the following activity. 3 Activity 1. Getting to Know You, Ellipse! Start getting to know what an ellipse is and its parts by getting a ruler and filling out the needed information below: Points F and G are called foci. A. Measure the distances between the given points on the ellipse in centimeters. 1. AF = _____ 2. BF = _____ 3. EF = _____ 4. CF = _____ AG = _____ BG = _____ EG = _____ CG = _____ AF + AG = _____ BF + BG = _____ EF + EG = _____ CF + CG = _____ 5. Identify any point on the ellipse. Name the point H. HF = _____ HG = _____ HF + HG = ______ B. What have you observed with the distances? ___________________________________________________________________________ ____________________________________________. C. Define an ellipse with your own words using what you have observed. ___________________________________________________________________________ ____________________________________________. 4 DISCOVER The ellipse is one of the conic sections. If you cut a double right circular cone with a plane in a slanted manner without passing through any of the bases, you will be able to make an ellipse. Perspective views Top view From Activity 1, you have learned the definition of an ellipse. Ellipse is a set of points whose distances from two fixed points called the foci add up to a certain constant. An ellipse when plotted on a rectangular coordinate system have two orientations. The horizontal ellipse and the vertical ellipse. Using a horizontal ellipse, the parts are the following: y W1 a c V1 C F1 F2 V2 x b W2 Center: Origin (0, 0) Foci: F1(-c, 0) and F2 (c, 0) The distance from the center to any of the foci is c units. For any point of the ellipse, the sum of its distances to the foci is 2a Vertices: V1(-a, 0) and V2(a, 0) The distance between the center to any of the vertices is a units. Points on the ellipse collinear with the foci and the center. The distance between V1 to V2 is equal to 2a. This segment is called major axis. The foci and the vertices is on the major axis. Covertices: W1(0, b) and W2(0, -b) The distance between the center to any of the Covertices is b units. The distance between W1 to W2 is 2b. This segment is called minor axis. 5 Using the parts of an Ellipse you are now ready to learn about its standard equations. There are four equations you need to learn. Each equation caters to a different orientation of the ellipse and the location of its center. Standard Equations of Ellipse with Center at the Origin: Ellipse Equation Horizontal with center at (0,0): Clue: Notice that a, half of the major axis is under the variable x (the horizontal axis). Vertical with center at (0,0): Clue: Notice that a, half of the major axis is under the variable y (the vertical axis). Wherein: Example 1: What is the standard equation of an ellipse with center at the origin, foci at (±4, 0) and vertices at (±6, 0)? Steps: 1. Sketch the ellipse and identify the orientation of the given ellipse. Identify the appropriate standard equation to use for substitution of values. Given: Foci at (4,0) and (-4,0) Vertices at (6,0) and (-6,0) V1 V2 F2 F1 Needed Equation: 2. Identify the value a and c then solve for the value of b. a = 6 units and c = 4 units -> -> -> -> 3. Substitute the value of a and b to the equation. 6 b= units Answer Example 2: Determine the standard equation of an ellipse with foci at (0, ±3) and whose any point on it, the sum of its distances from the foci is 10. Steps: 1. Sketch the ellipse. Identify the orientation and identify the appropriate equation to use. Note: The foci are on the major axis and the foci is located at (0,3) and (0,-3). F1 Needed Equation: F2 2. Identify the value of a and c and solve for the value of b. Note: The sum of distances is equal to 2a 2a = 10 units a = 5 units c = 3 units -> 3. Substitute the value of a and b to the equation. -> b = 4 units -> -> Answer On your own, try finding the standard equation of the following ellipses: Ellipse 1. Foci at (±2,0) and a Vertex at (-3.5,0). Ellipse 2. Foci at (0, ±7) and whose any point on it, the sum of its distances is 16. Ellipse 3. Center at the origin, a focus at (-5.5,0) and a vertex at (-6.5,0). After solving, check if your answer is the same as the correct one below: I hope you got the same answers! Ellipse 1: Ellipse 2: Ellipse 3: 7 Standard Equations of Ellipse with Center at (h,k): Note: (h,k) is the coordinate of the center of the ellipse. Ellipse Equation Horizontal with center at (h,k): Clue: Notice that h is always with x and k is always with y. a is still under x if the ellipse is horizontal and a is under y if the ellipse is vertical. Vertical with center at (h,k): Wherein: Example 3. Determine the standard equation of an ellipse with foci at (-3,0) and (-3,10) and vertices at (-3,-2) and (-3,12). Steps: 1. Sketch the ellipse and identify the orientation of the ellipse. Foci at (-3,0) and (-3,10) Vertices at (-3,-2) and (-3,12) The sketch revealed a vertical ellipse. You can also find the orientation of the ellipse through inspection. From the given coordinates of the foci and vertices, notice that only the y-coordinate varies in value. (y the vertical axis) Needed Equation: 8 2. Identify the coordinate of the center of the ellipse and the value of a, c and b. For the coordinate of the center, (h,k): Solve for the midpoint of the foci, (-3,0) and (-3,10). Note: the center is the midpoint of the foci and also the midpoint of the vertices. h= k= (h,k) = (-3, 5) For a: Note: it is the distance from vertex to center One vertex is at (-3,-2) and center is at (-3, 5) c = 5 units -> 3. Substitute the value of h, k, a and b to the equation. -> b= units -> t Answer Example 4. Find the standard equation of an ellipse with one focus at (-3.5,6), vertex at (-5,6) and center at (-1,6). Steps: 1. Identify the orientation of the ellipse and decide on the appropriate standard equation to utilize. Through inspecting the given, (-3.5,6), (-5,6) and (-1,6) the value of the xcoordinate varies. This means the ellipse is horizontal (x, the horizontal axis). You can use this technique or you can sketch the ellipse. Needed Equation: 2. Identify the value of a, c and b. For a: Use coordinate of center and vertex. (Or you can count the distance between them with the use of your sketch of the ellipse.) C(-1,6) and V(-5,6). The absolute value of the difference between -1 and -5 is 4 units. a = |-1-(-5)| = 4 units a = 4 units For c: Use the coordinate of center and the foci. (Or you can count the distance between them with the use of your sketch of the ellipse.) C(-1,6) and F(-3.5,6). The absolute value of the difference between -1 and -3.5 is 2.5 units. c = |-1-(-3.5)| = 2.5 units 9 For b: c = 2.5 units -> t -> b= 3. Substitute the value of h, k, a and b to the equation. -> t t t Answer On your own, try finding the standard equation of the following ellipses: Ellipse 4. Center at (-5,1), a focus at (-2,1) and a vertex at (-1,1). Ellipse 5. Foci at (0, -7) and (0,3) and Vertices at (0,5) and (0,-9). Ellipse 6. Foci at (-2,0) and (4,0) and whose any point on it, the sum of its distances are 8. After solving, check if your answer is the same as the correct one below: I hope you got the same answers! Ellipse 4: Ellipse 5: Ellipse 6: t 10 units EXPLORE Here are some enrichment activities for you to work on to master and strengthen the basic concepts you have learned from this lesson. Activity 1. Standard Equation of the Ellipse Determine the standard equation of the given ellipses. 1. Ellipse A 2. Ellipse B 11 Activity 2. Standard Equation of the Ellipse You Illustrated Find the standard equation of the following ellipse that satisfies the given conditions: 1. Vertices at (0, ±3) and foci at (0, ±2). 2. One vertex at (4, 3), foci at (-3, 3) and (3, 3). Enrichment Activity 1. A Planet’s Elliptical Path! A planet called Diwata follows an elliptical orbit. The Diwata’s orbit is drawn down to scale by a certain Filipino Astronomer. Through observation, he was able to compute for the vertices and measure the distance between the planet and the sun, one of its foci. What you need: Ruler Pen/ pencil What you have to do: Utilize the given figure. a. Use your ruler and make a line connecting the two vertices. b. Use your ruler and find the value of 2a by measuring the distance between the two vertices. (Use centimeters) c. Identify the center by measuring half the value of 2a from one vertex. Assume that the center is at (0,0). d. Measure the distance from the foci to the center. SCALE: 1 cm: 430 million kilometers Figure 2. Elliptical Orbit of Planet Diwata Assessment 1. Direction: Answer the following questions. All answers should be in centimeters. 1. Distance between the two vertices: _________________. 2. Measure of a: ___________ ; Measure of c: ___________ 3. Computed value of b: ____________ 4. Standard Equation of the Elliptical orbit of planet Diwata: _________________ 12 Enrichment Activity 2. Locate are the Foci! A whispering gallery is to be drawn by an Architect who was only given two dimension by the owner. It should be 96 feet long and 46 feet wide in elliptical shape. 1. What is the standard form of the equation of the ellipse representing the outline of the room? Hint: assume a horizontal ellipse, and let the center of the room be the point (0,0). 2. How far apart should the two foci be? Given: Solution: Answers: Great job! You have understood the lesson! Let’s apply what you have learned in a tangible output. 13 DEEPEN Make your own ellipse using the definition of ellipse and then determine the standard equation of the ellipse you made. Prepare the following material and then follow the given instruction. Let’s make an ellipse! Materials: - string - ballpen - clean cardboard (this will be submitted along with the module) - pushpins/thumbtacks - corkboard or any hard surface placed on the table Directions: 1. Mount the cardboard on the hard surface using the pushpins. 2. Cut the string long as the size of your ballpen. 3. Pin each end of the string on the cardboard in such a way that the distance between the two pins is a little bit smaller than the size of the ballpen. *Make sure to knot the string on the push pin before pinning it on the cardboard. 4. Remove the cap of you ballpen and put the tip of the ballpen on the side of the string. 5. Move the ballpen left to right by pushing away from the center using the string to pull the tip of the ballpen. *Adjust the position of the ballpen to be able to complete the shape. See figure 3 below: Figure 3 6. Remove the pins and the string. Draw a line connecting the two Foci (the location of the pins using the ruler. 7. Measure the necessary parts of the ellipse using the ruler (use centimeters) and determine the standard equation of the ellipse you have made. 8. Use the same cardboard to write all the needed information, your solution and the standard equation. 14 Rubrics for Scoring the Output Component Poor 1 point Drawn Ellipse The drawn ellipse is not well defined. Technical Line and Points Measurements Solution Over-all presentation The necessary line and points are not properly drawn and are incorrectly located. All values are incorrectly measured. The solution is incorrect and done in a nonsensical manner. The output is dirty and presented in a disorderly manner. 15 Good 3 points The drawn ellipse is defined except for some of its parts. Very Good 5 points The drawn ellipse is very well defined. Some necessary line and points are properly drawn and correctly located. All the necessary line and points are correctly and precisely drawn and located. Some values are correctly measured. All values are correctly measured. Some parts of the solution is correct and done in a logical manner. The solution is correct and done in a logical manner. The output is somewhat neat and presented in an almost organized manner. The output is neat and presented in an organized manner. GAUGE Direction: Write the letter of the correct answer on the blank provided before the number. ______ 1. What is the definition of an ellipse? A. A set of points whose distance from a fixed point called the center is a certain constant. B. A set of points whose distances from two fixed points called the foci add up to a certain constant. C. A set of points whose distances from two fixed points called the foci differs to a certain constant. D. A set of points whose distances from a fixed point called the focus and a fixed line called the directrix is a certain constant. ______ 2. Which among the following cases shows a horizontal ellipse? A. One focus at (-1,-1) center at (-1,1) B. Vertices located at (±5, 0), foci at (±3,0) C. Vertices located at (0, ±5), foci at (0, ±3) D. Covertices located at (±4,0), foci at (0, ±3) ______ 3. What is the standard equation of a horizontal ellipse with center at the origin? A. C. B. D. ______ 4. What is the standard equation of a vertical ellipse with center at the origin? A. C. B. D. ______ 5. What is the standard equation of a horizontal ellipse with center at the point (h, k)? t A. t C. B. t t D. ______ 6. What is the standard equation of a vertical ellipse with center at the point (h, k)? A. t t C. B. D. 16 t t ______ 7. What is the standard equation of an ellipse with foci at (±4,0); length of major axis 10? A. B. t t t t t C. D. t t t t t ______ 8. What is the standard equation of an ellipse with center at (0,0), focus at (0,3) and vertex at (0,5)? A. B. t t + + t t t =1 C. =1 D. t t t t + + =1 t =1 ______ 9. What is the standard equation of an ellipse with center at (-1,0), a vertex at (-4,0) and a focus at (-3,0)? A. B. t t + + t t =1 C. =1 D. t - t - t =1 t =1 - t ______ 10. What is the standard equation of an ellipse with center at (-2,2), a vertex at (-2,4), a focus at (-2,3)? A. B. t t + + t t =1 C. =1 D. 17 t t - t =1 =1 REFERENCES Printed Materials Department of Education Republic of the Philippines. (2016). Precalculus Learner’s Material (First Edition). Sunshine Interlinks Publishing House, Inc.. Quezon City, Philippines Department of Education Republic of the Philippines. (2016). Precalculus Teacher’s Guide (First Edition). Sunshine Interlinks Publishing House, Inc.. Quezon City, Philippines Comandante, Jr. F. L. (2000). Analytic and Solid Geometry Made Easy. National Bookstore. Mandaluyong City, Philippines Websites Kepler’s Law from Universe Today Space and Astronomy News. (2010). Retrieved July 24, 2020 from https://www.universetoday.com/55423/keplers-law/ Image of Conic Sections. Retrieved July 24, 2020 from https://upload.wikimedia.org/wikipedia/commons/9/97/Conic_section_%2 8PSF%29.png Elements of Astronomy from Interactive Book Image page 163. Retrieved July 24, 2020 from https://www.flickr.com/photos/internetarchivebookimages/14598208288/ Solving Applied Problems Involving Ellipses by Lumen College Algebra Ellipses. Retrieved July 25, 2020 from https://courses.lumenlearning.com/ivytechcollegealgebra/chapter/solving-applied-problems-involving-ellipses/ 18 KEY ANSWERS PRE-TEST AND GAUGE 1. B 2. B 3. A 4. B 5. A 6. B 7. A 8. B 9. B 10. A Activity 1. Getting to Know You, Ellipse! A. Measure the distances between the given points on the ellipse in centimeters. 1. AF = 4.1cm_ 2. BF = 9.4cm 3. EF = 3.2cm 4. CF = 8.0cm AG = 8.5cm BG = 3.2cm EG = 9.4cm CG = 4.6cm AF + AG = 12.6cm BF + BG = 12.6cm EF + EG = 12.6cm CF + CG = 12.6cm 5. Identify any point on the ellipse. Name the point H. HF = _____ HG = _____ HF + HG = 12.6cm B. What have you observe with the distances? __The sum of the distances from the two foci is constant____________________. C. Define an ellipse with your own words using what you have observed. ___A set of points whose distances on the foci is constant____________________. Activity 1. Standard Equation of the Ellipse 1. 2. t t + t + =1 t =1 Activity 2. Standard Equation of the Ellipse You Illustrated 1. 2. t + t t + t t =1 =1 Enrichment Activity 1. A Planet’s Elliptical Path! Assessment 1. Direction: Answer the following questions. All answers should be in centimeters. 1. Distance between the two vertices: ___11.2 cm______. 2. Measure of a: __5.6cm___ ; Measure of c: __3.5cm___ 3. Computed value of b: ___19.11cm___ Standard Equation of the Elliptical orbit of planet Diwata: 19 t t + t =1 a Enrichment Activity 2. Locate are the Foci! Given: 2a = 96 feet a = 48 feet 2b = 46 feet b = 23 feet Solution: 1. t + t =1 -> t + t =1 2. c2 = a2 – b2 -> c2 = 2304 – 529 = 1775 -> c = = 42.17 feet distance between the two foci: 2c = 2(42.17 feet) = 84.26 feet 20
