Submitted By ilovemeame

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Words 520

Pages 3

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, A and C are not equal but of the same sign. Thus,the general equation of the ellipse is Ax2 + Cy2 + Dx + Ey + F = 0 or

Standard Equations of Ellipse

From the figure above, and

From the definition above,

Square both sides

Square again both sides

From triangle OV3F2 (see figure above)

Thus,

Divide both sides by a2b2

The above equation is the standard equation of the ellipse with center at the origin and major axis on the x-axis as shown in the figure above. Below are the four standard equations of the ellipse. The first equation is the one we derived above.

Ellipse with center at the origin

Ellipse with center at the origin and major axis on the x-axis.

Ellipse with center at the origin and major axis on the y-axis.

Ellipse with center at (h, k)

Ellipse with center at (h, k) and major axis parallel to the x-axis.

Ellipse with center at (h, k) and major axis parallel to the y-axis.

The Hyperbola

Submitted by Romel Verterra on February 21, 2011 - 1:33pm

Definition

Hyperbola can be defined as the locus of point that moves such that the difference of its distances from two fixed points called the foci is constant. The constant difference is the length of the transverse axis, 2a. General Equation

From the general equation of any conic (A and C have opposite sign, and can be A > C, A = C, or A < C.) or

Standard Equations

From the definition:

From the figure:

Thus,

The equation we just derived above is the standard equation of hyperbola with center…...

...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. 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. KEPLER’S LAWS OF......

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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 _________________________________________ Question 2) Given the equation of the parabola answer the following questions. (10 marks) a) b) c) i. What is the vertex? ii. Does the parabola face up or down? iii. Is the parabola wide or narrow compared to the basic parabola? iv. What is the axis of symmetry? v. Make an x-y table to show at least 4 coordinate points on the parabola. vi. Graph the parabola showing the......

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...recognition and mapping, so they can be a part of the so called “Driver Support Systems”. The DSS is in a way the predecessor of the car autopilot. 174 John Hatzidimos - Automatic traffic sign recognition in digital images Traffic sign recognition is part of the general case of Pattern Recognition. Major problem in pattern recognition is the difficulty of constructing characteristic patterns (templates). This is because of the large variety of the features being searched in the images, such as people faces, cars, etc.. On the contrary, traffic signs a) are made with vivid and specific colors so as to attract the driver’s attention and to be distinguished from the environment b) are of specific geometrical shapes (triangle, rectangle, circle-ellipse) and c) for each sign there is a specific template. It is therefore rather easy to develop an algorithm in such a way that the computer has “a priori knowledge” of the objects being searched in the image. The developed algorithm is divided in two basic phases each one composed of a certain number of steps. In the first phase the detection of the location of the sign center of gravity (which is used as a location reference point) in the image coordinate system is carried out, based on its geometric characteristics. The second phase is the sign recognition with the matching between the search image and the template images, already stored in a database. The programming language used is IDL (Interactive Development Language), of......

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...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......

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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......

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...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 x^2/a^2 + y^2/b^2 = 1 for horizontal ellipse, a>b and for vertical, b>a. The equation of the hyperbola is x^2/a^2 – y^2/b^2 = 1. I believe that the professor should focus on the difference between the equation of the ellipse and the equation of the hyperbola because not a lot of people in the classroom surely understand the differences and how to find the vertices and the foci of the ellipse and hyperbola. Actually I need a little bit understanding on this material for a test so I can have a clearer understanding of it. I just need a review on the material and I will fully understand it. I believe that the instructor is doing the best......

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...Business Impact Analysis (BIA) Preliminary System Information |Omnitrans Dept. Omnitrans |Date BIA Completed: 9/12/2006 | |System Name: Ellipse |BIA POC: Don Walker | |System Function: Payroll | |System Manager Point of Contact (POC): Don Walker | |System Description: {Discussion of the system purpose and architecture, including system diagrams} Omnitrans payroll time | |calculations’, direct deposit process, and check printing. | | | |A. Identify System POCs Role | |Internal {Identify the individuals, positions, or offices within your organization that depend on or support the system; also | |specify their relationship to the system} | | | ...

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...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 terms have the opposite signs. If so then its a hyperbola, if not move on to see if the squared terms are multiplied by the same number. If so it’s a circle if not then its an ellipse. APPLICATIONS OF A CONIC SECTION Circle: x2 + y2 + Dx + Ey + F = 0 Ellipse: Ax2 + Cy2 + Dx + Ey + F......

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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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... Apostrophe, Quotation Marks and Ellipses The final three punctuation forms in English grammar are the apostrophe, quotation marks and ellipses. Unlike previously mentioned grammatical marks, they are not related to one another in any form. An apostrophe (') is used to indicate the omission of a letter or letters from a word, the possessive case, or the plurals of lowercase letters. Examples of the apostrophe in use include: • Omission of letters from a word: An issue of nat'l importance. • Possesive case: Sara's dog bites. • Plural for lowercase letters: Six people were told to mind their p's and q's. It should be noted that, according to Purdue University, some teachers and editors enlarge the scope of the use of apostrophe, and prefer their use on symbols (&'s), numbers (7's) and capitalized letters (Q&A's), even though they are not necessary. Quotations marks ( “” ) are a pair of punctuation marks used primarily to mark the beginning and end of a passage attributed to another and repeated word for word. They are also used to indicate meanings and to indicate the unusual or dubious status of a word. Single quotation marks (') are used most frequently for quotes within quotes. The ellipses mark is generally represented by three periods (. . . ) although it is occasionally demonstrated with three asterisks (***). The ellipses are used in writing or printing to indicate an omission, especially of letters or words. Ellipses are frequently used within......

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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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...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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...d1= a+4 b 3 3 – 2;d b = d1 – d d1 = b + d 4 The equation for the M1 is d1 = b+d M2 : d2 – Equation for the M2 midpoint d1 = a(x+3/2) + b(y+2) + c d1 = ax + by + c +3 a/2 +2 b d1 = 0+ 3a/2 + 2b ×2; d2 = 3a +4b 5 5 – 2; a + b = d2 – d d2 = a + b + d 6 The equation of the M2 midpoint is d1 = a + b + d As mentioned above the algorithm chooses between N and NE pixels on the sign of the decision variable calculated in previous method. * Algorithm for the ellipse The equation for the ellipse is : x2/a2 + y2/b2 = 1 x2/a2 + y2/b2 – 1 = 0 x2 b2 + y2 a2 - a2 b2 = 0 Center of the ellipse (x,y) = (0,0) a = rx , b = ry The first point is considered from the origin (x,y) = (0,b) The ellipse has 4 -fold symmetry, so that we have to consider ¼ of the ellipse to implement the algorithm. When we consider about the first region of the ellipse, m<0 In region 2 of the ellipse, m>0 Therefore we must find the point which changes the direction. Direction Changing Criterion Scenario 1 - Region 1: ∂f(x,y)∂x - ∂f(x,y)∂y< 0 2b2x - 2a2y< 0 b2x - a2y< 0 ÷2, b2x < a2y In this region, units steps at x direction. In the first scenario we can choose East pixel and SouthEast pixels d = initial decision variable. d = X2b2 + Y2a2 - a2b2 d= (x+1)2b2 +(y-1/2)2 - a2b2 d = x2b2 + y2a2 - a2b2 + 2b2x + b2 - a2y + a2/4 d = 0 + 2b2x + b2 - a2y + a2/4 ;Since at......

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...Intelligent Fresh Engaging Unexpected Too abstract Overly clever Superficial Relevant Contemporary Chic Elegant Simple Clean Just a fad Too bold 14 New brand platform New brand platform 15 Interpretation of brand personalities Interpretation of brand personalities Optimistic Inviting Samsung always sees possibilities. Samsung is magnetic; it captivates with an inclusive attitude. IT is … IT is not … IT is … IT is not … Inspiring Bright Upbeat Too far-fetched Unrealistic Engaging Charismatic Forward-thinking Pretentious Exclusionary Aloof Ordinary 16 New brand platform New brand platform 17 Bringing the brand to life 20 Brand story 51 Ellipse 25 Campaign idea 73 Imagery 26 Look, tone, and manner of the campaign 83 Copy elements 27 Campaign do’s and don’ts 28 Visual elements 31 Wordmark 35 Typography 45 Color Building our brand story Building an authentic Samsung brand story In order to deliver the Samsung brand equity to consumers in an ownable, differentiated way, we need to translate that brand equity into a brand story for communication to Young-Minded Consumers. Creating a powerful, big idea — Imagination Global brands need big ideas that tap into universal human emotions and link consumer mindsets across markets. Imagination is one such universal and aspirational consumer truth, particularly attractive to......

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... 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 parabola. Finally, the plane may have a smaller angle to the vertical axis (that is, the plane is steeper than the generator), in which case the plane will cut both nappes of the cone. The resulting curve is called a hyperbola, and has two disjoint “branches.” Notice that if the plane is actually perpendicular to the axis (that is, it is horizontal) then we get a circle – showing that a circle is really a special kind of ellipse. Also, if the intersecting plane passes through the vertex then we get the so-called degenerate conics; a single point in the case of an ellipse, a line in the case of a parabola, and two intersecting lines in the case of a hyperbola. Although intuitively and visually appealing, these definitions for the conic sections tell us little about their properties and uses. Consequently, one should master their “plane geometry” definitions as well. It is from these definitions that their algebraic......

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