...Definition of Ellipse Ellipse is the locus of point that moves such that the sum of its distances from two fixed points called the foci is constant. The constant sum is the length of the major axis, 2a. General Equation of the Ellipse From the general equation of all conic sections, [pic] and [pic] are not equal but of the same sign. Thus, the general equation of the ellipse is [pic] or [pic] Standard Equations of Ellipse Elements of the ellipse are shown in the figure above. 1. Center (h, k). At the origin, (h, k) is (0, 0). 2. Semi-major axis = a and semi-minor axis = b. 3. Location of foci c, with respect to the center of ellipse. [pic]. 4. Length latus rectum, LR 5. Consider the right triangle F1QF2: Based on the definition of ellipse: [pic] [pic] [pic] By Pythagorean Theorem [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] You can also find the same formula for the length of latus rectum of ellipse by using the definition of eccentricity. 6. Eccentricity, e DEFINITION: Eccentricity of Conic Eccentricity is a measure of...
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...generator will sweep out a surface, as shown in the diagram. It is this surface which we call a cone. Notice that a cone has an upper half and a lower half (called the nappes), and that these are joined at a single point, called the vertex. Notice also that the nappes extend indefinitely far both upwards and downwards. A cone is thus completely determined by its vertex angle. Now, in intersecting a flat plane with a cone, we have three choices, depending on the angle the plane makes to the vertical axis of the cone. First, we may choose our plane to have a greater angle to the vertical than does the generator of the cone, in which case the plane must cut right through one of the nappes. This results in a closed curve called an ellipse. Second, our plane may have exactly the same angle to the vertical axis as the generator of the cone, so that it is parallel to the side of the cone. The resulting open curve is called a...
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...and all of your steps. (Hint: Use the properties of logarithms.) (4 marks each) a) b) c) d) Question 6) Solve for the variable. Show all of your work and all of your steps. Show the answer to 4 decimal places. (Hint: Use the common logarithm.) (4 marks each) a) b) c) Question 7) Solve for . Show all of your work and all of your steps. Show the answer to 4 decimal places. (Hint: Use the natural logarithm and the definition of a logarithm.) (4 marks each) a) b) c) Question 8) Ms. Mary bought a condo for $145 000. Assuming that the value of the condo will appreciate at most 5% a year, how much will the condo be worth in 5 years? Section 2: Conic Sections Standard forms to Know: * Parabola * Circle * Ellipse * And what does a hyperbola look like? (No formula necessary) Question 1) Write an equation for the circle that satisfies each set of conditions. (2 marks each) a) centre (12, -4), radius 81 units _________________________________________ b) centre (0, 0), radius 3/5 units...
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...college algebra problems on the equation of ellipses are presented. Problems on ellipses with detailed solutions are included in this site. The solutions are at the bottom of the page. 1. What is the major axis and its length for the following ellipse? (1/9) x 2 + (9/25) y 2 = 1/25 2. An ellipse is given by the equation 8x 2 + 2y 2 = 32 . Find a) the major axis and the minor axis of the ellipse and their lengths, b) the vertices of the ellipse, c) and the foci of this ellipse. 3. Find the equation of the ellipse whose center is the origin of the axes and has a focus at (0 , -4) and a vertex at (0 , -6). 4. Find the equation of the ellipse whose foci are at (0 , -5) and (0 , 5) and the length of its major axis is 14. 5. An ellipse has the x axis as the major axis with a length of 10 and the origin as the center. Find the equation of this ellipse if the point (3 , 16/5) lies on its graph. 6. An ellipse has the following equation 0.2x 2 + 0.6y 2 = 0.2 . a) Find the equation of part of the graph of the given ellipse that is to the left of the y axis. b) Find the equation of part of the graph of the given ellipse that is below the x axis. 7. An ellipse is given by the equation (x - 1) 2 / 9 + (y + 4) 2 / 16 = 1 . Find a) its center, a) its major and minor axes and their lengths, b) its vertices, c) and the foci. 8. Find the equation of the ellipse whose foci are at (-1 , 0) and (3 , 0) and...
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...MA1200 Basic Calculus and Linear Algebra I Lecture Note 1 Coordinate Geometry and Conic Sections υ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections Topic Covered • Two representations of coordinate systems: Cartesian coordinates [ሺݕ ,ݔሻcoordinates] and Polar coordinates [ሺߠ ,ݎሻ-coordinates]. • Conic Sections: Circle, Ellipse, Parabola and Hyperbola. • Classify the conic section in 2-D plane General equation of conic section Identify the conic section in 2-D plane - Useful technique: Rotation of Axes - General results φ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections Representations of coordinate systems in 2-D There are two different types of coordinate systems used in locating the position of a point in 2-D. First representation: Cartesian coordinates We describe the position of a given point by considering the (directed) distance between the point and -ݔaxis and the distance between the point and -ݕaxis. ݕ 0 ܽ ܲ ൌ ሺܽ, ܾሻ ܾ ݔ Here, ܽ is called “-ݔcoordinate” of ܲ and ܾ is called “-ݕcoordinate” of ܲ. χ MA1200 Basic Calculus and Linear Algebra I Lecture Note 1: Coordinate Geometry and Conic Sections ܲଶ ൌ ሺݔଶ , ݕଶ ሻ ܲଵ ൌ ሺݔଵ , ݕଵ ሻ Given two points ܲଵ ൌ ሺݔଵ , ݕଵ ሻ and ܲଶ ൌ ሺݔଶ , ݕଶ ሻ, we learned that • the distance between ܲଵ and ܲଶ : ܲଵ ܲଶ ൌ ඥሺݔଶ െ ݔଵ ሻଶ ሺݕଶ െ ݕଵ...
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...with center at and radius 2. Write the equation in standard form for the circle with center at and radius 3. Graph the circle given by 4. Graph the circle given by 5. Write the equation of the circle in standard form given: 9.1 Parabolas: 6. Find the focus of the parabola 7. Find the focus of the parabola 8. Write the equation of the parabola in standard form and find the focus and directrix. 9. Write the equation of the parabola in standard form and find the focus and directrix. 10. Write the equation for the parabola with vertex and focus 11. Write the equation for the parabola with vertex and directrix 9.2 Ellipses: 12. Identify the center, vertices, & foci of the ellipse given by and graph. 13. Identify the center, vertices, & foci of the ellipse given by and graph. 14. Write the equation in standard form: 9.5 Parametric Equations: 15. Write and in rectangular form. 16. Write each pair of parametric equations in rectangular form: 17. Write and in rectangular form. 18. Write and in rectangular form. 9.6 Polar Equations: 19. Graph the following polar coordinate: 20. Graph the following polar coordinate: 21. Graph the following polar coordinate: 22. Graph the following polar coordinate: 23. Find the polar coordinate of the point 24. Find the polar coordinate of the point 25. Find the polar coordinate of the point 26. Find the rectangular...
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...through the cone you are actually taking a section of the cone and from that section you are dealing with a shape. If you take an exact slice through the cone horizontally, you are left with a circle. If you take a slice roughly at a forty five degree, you will be dealing with an ellipse. If you take a slice that is parallel from one edge of the cone to the other cone, you are dealing with a parabola. If you take a slice from directly off centered but straight down from top to bottom, you give yourself a hyperbola. These are a few terms with definitions you will see while working with conic sections. In a circle, ellipse, and a hyperbola you have a Center. Which is usually at the point of (h,k.) The focus or “Foci” is the point which distances are measured in forming the conic. The directrix is the distance that is measured in forming the conic. The major access is the line that is perpendicular to the directrix that passes through the foci. Half of the major axis between the center and the vertex is called the semi major access. There is a general equation that covers all the conic sections and goes as follows: Ax2+Bxy+Cy2 + Dx+Ey+F=0. From this equation you can create equations for circles, ellipses, parabolas and hyperbolas. There is a test to find out which conic section you are dealing with by just looking at the equations. If both variables not squared then it’s a parabola, if it is you can move on and look to see if the squared...
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...* Link Mechanism A link mechanism can be defined as a system of connecting parts/rods that move or work together when a particular action is pre-determined. * Parabola The path traced out by a point, which moves in a plane, so that the ratio of its distance from a fixed point [FOCUS] to any point on the curve, and from the curve to a perpendicular distance on the directrix is always constant and equal to one. This constant is also known as the eccentricity. * Ellipse The path traced out by a point, which moves in a plane, so that the ratio of its distance from a fixed point [FOCUS] to any point on the curve, and from the curve to a perpendicular distance on the directrix is always constant and less than one. This constant is also known as the eccentricity. * Hyperbola: The path traced out by a point, which moves in a plane, so that the ratio of its distance from a fixed point [FOCUS] to any point on the curve, and from the curve to a perpendicular distance on the directrix is always constant and greater than one. This constant is also known as the eccentricity. * Archimedean Spiral The path traced out by a point along a rod, as the rod pivots about a fixed end. The linear movement of the point along the rod is constant with the angular movement of the rod. * Involute The path traced out by a point when an end of a plane figure is wrapped or unwrapped when held firm. * Epicycloid Tracing the path of a point as a circular disc rolls on the outside...
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...Hypatia, regularly called Hypatia of Alexandria, was a Greek mathematician, astronomer, and philosopher in Egypt, then a part of the Eastern Roman Empire. Her birthplace was in Alexandria, Egypt. Hypatia was born in 370 CE and she was the daughter of the mathematician Theon. Later on, after a few years she passed away on March 415 CE. As she passed away, she died at the age of sixty years old. While the mathematician Hypatia, enjoyed her life, she sometimes experienced some historical world events during her life. For instance, as stated by users.ox.ac.uk, it said “ The great library at Alexandria was founded by Ptolemy I at the end of the fourth century BCE. It was said to be the largest collection of books in the ancient world (over half...
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...1. Since we are looking for the time it takes for the student to be able to type 55 WPM N(t) will equal 55, when we plug this into the equation and solve, we see that the it would take approximately 8 weeks for the student to type 55 WPM 55 = 157/(1+5e^-0.12t) 1+5e^-0.12t = 157/55 ≈ 2.85454 5e^-0.12t = 1.85454 e^-0.12t = 1.85454/5 = 0.370908 -0.12t = ln 0.370908 t = ln 0.370908/-0.12 = 8.265 ≈ 8 weeks 2. According to the information (bacteria in culture after 3 hours is 100 and 5 hours is 400), the bacteria appears doubling every hour. If this is the case a) the growth rate is 200% every hour . b) Following this same trend to find the initial growth rate we would divide 100 by 2, 3 times. Thus the initial number of bacteria would be 12.5 c) Since the number of bacteria of 5 hours is 400, to find the number of bacteria at 6 hours is going to be double the amount at 5 hours, 800. 3. To solve we need to find the equation of the parabola in vertex form, since we know a point of the vertex, also that is vertically symmetrical from this point. Recall that the equation we’ll use is the quadratic y = a(x-h)² + k. Since we are looking for how wide the parabola is at 10 meters(8 meters) and its vertex’s x coordinate is at 0 and its vertically symmetrical the a point that is passed through at that height(10 meters) is 4 (4,10).Substitute this information into the equation (10=a(4-0)^2) + 12 simplify -2=16a 12/16=a a = -1/8 Now substitute the equation with a=-1/8...
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...Research on Butterfly Theorem Butterfly Theorem is one of the most appealing problems in the classic Euclidean plane geometry. The name of Butterfly Theorem is named very straightforward that the figure of Theorem just likes a butterfly. Over the last two hundreds, there are lots of research achievements about Butterfly Theorem that arouses many different mathematicians’ interests. Until now, there are more than sixty proofs of the Butterfly Theorem, including the synthetical proof, area proof, trigonometric proof, analytic proof and so on. And based on the extension and evolution of the Butterfly Theorem, people can get various interesting and beautiful results. The definition of the Butterfly Theorem is here below: “Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD cuts PQ at X and BC cuts PQ at Y. Prove that M is also the midpoint of XY.” (Bogomolny) This is the most accurate definition currently. However, Butterfly Theorem has experienced some changes and developments. The first statement of the Butterfly Theorem appeared in the early 17th century. In 1803, a Scottish mathematician, William Wallace, posed the problem of the Butterfly Theorem in the magazine The Gentlemen’s Mathematical Companion. Here is the original problem below: “If from any two points B, E, in the circumference of a circle given in magnitude and position two right lines BCA, EDA, be drawn cutting the circle in C and D, and meeting in A; and...
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...Brand guide Version 1.0 This book presents a new brand strategy for Samsung: — who we serve, — what we stand for, and — how we communicate our value. It begins by painting a clearer picture of our core consumer, then defines a new brand platform that will help us build a more powerful emotional connection with this target. Finally, it provides the visual and verbal elements we need to bring our brand story to life. Think of this book as a user’s manual for our brand. It will help all of us make Samsung a more powerful global icon. This document is intended for Samsung internal purposes only. The information contained herein is proprietary and confidential. Any use, copying, retention or disclosure by any person other than the intended recipient or the intended recipient’s designees is strictly prohibited. © 2008, Samsung Electronics Co. Table of contents 1 New brand platform 19 Bringing the brand to life 93 Applications 121 Appendix New brand platform 2 Global brand objective 3 Brand target 4 Target profiles 10 Brand equity pyramid 12 Brand equity 14 Interpretation of brand personalities Global brand objective The Samsung brand has come a long way in a short time. Our first focus was to build brand awareness worldwide. We succeeded by making Samsung one of the best known brands in any category. In the following years, our task was to build our premium quality, to help drive preference against competing...
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...things I’ve had learned from this course and one of them was that we use Math for everyday life. I’ve also learned many ways how to solve equations such as linear, quadratic, exponential, and logarithmic equations. All the material that we did learn was all easy to learn and understand. I believe that the instructor did a good job explaining on how to solve problems. If my friend was asking me how to determine the differences between the equation of the ellipse and the equation of the hyperbola, I would first give he or she the definition of the two words ellipse and hyperbola. An ellipse is a set of all points in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci) A hyperbola is the set of all points in a plane for which the absolute value of the difference of the distances from two distinct fixed points called foci is constant. The equations for ellipse and hyperbola are different. You write them in standard form. The equation of the ellipse is...
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...motion and universal gravitation were true. Developed calculus KEPLER’S LAWS OF PLANETARY MOTION 1. Planets move around the Sun in ellipses, with the Sun at one focus. 2. The line connecting the Sun to a planet sweeps equal areas in equal times. 3. The square of the orbital period of a planet is proportional to the cube of the semimajor axis of the ellipse. INITIAL VALUES AND EQUATIONS Unit vectors of polar coordinates (1) INITIAL VALUES AND EQUATIONS From (1), (2) Differentiate with respect to time t (3) INITIAL VALUES AND EQUATIONS CONTINUED… Vectors follow the right-hand rule (8) INITIAL VALUES AND EQUATIONS CONTINUED… Force between the sun and a planet (9) Newton’s 2nd law of motion: F=ma (10) F-force G-universal gravitational constant M-mass of sun m-mass of planet r-radius from sun to planet INITIAL VALUES AND EQUATIONS CONTINUED… Planets accelerate toward the sun, and a is a scalar multiple of r. (11) INITIAL VALUES AND EQUATIONS CONTINUED… Derivative of (12) (11) and (12) together (13) INITIAL VALUES AND EQUATIONS CONTINUED… Integrates to a constant (14) INITIAL VALUES AND EQUATIONS CONTINUED… When t=0, 1. 2. 3. 4. 5. KEPLER’S LAWS OF PLANETARY MOTION 1. Planets move around the Sun in ellipses, with the Sun at one focus. 2....
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...------------------------------------------------- EXERCISE 5.2 (3.5 hours) Assessment Preparation Checklist To prepare for this assessment: * Read Section 10.1: The Ellipse, pp. 890–901, Section 10.2: The Hyperbola, pp. 902–916, and Section 10.3: The Parabola, pp.916–925 from your textbook, Algebra and Trigonometry. These topics will introduce you to the concepts such as hyperbola and parabola. * Review the Module 5 lesson. This lesson will provide you various examples of the topics covered in this module. Title: Graphing Ellipse, Hyperbola, and Parabola Answer the following questions to complete this exercise: 1. Find the standard form of the equation of the ellipse and give the location of its foci. The standard form of the equation of an ellipse with the center at the origin and major and minor axes of lengths 2a and 2b (where a and b are positive, and a2 > b2) is: The location of foci are at (–c, 0) and (c, 0) where c2 = a2 – b2. 2. Graph the ellipse. and choose the correct graph from the given graphs: a. b. c. d. [Hint: To graph this ellipse, find the center (h, k) by comparing the given equation with the standard form of equation centered at (h, k). Next, find a and b. Find the vertices (h – a, k) and (h + a, k). Find the foci (h + c, k) and (h – c, k).] 3. Find the standard form of the equation of the hyperbola whose graph is given below. 4. Find the vertices of the hyperbola. ...
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