ellipse (An Introduction)
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An ellipse is a curve where the sum of the distances from two fixed focal points is a constant. It is defined as the set of all points whose distances from two focal points add up to a constant. The standard equation of an ellipse is (x/a)^2 + (y/b)^2 = 1, where the focal points are located at (±c,0) and the vertices are located at (±a,0) and covertices at (0,±b). Examples are given of finding the focal points, vertices, and covertices of ellipses with given standard equations and of writing the standard equation of an ellipse given its focal points and constant sum of distances.
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The document discusses different types of conic sections including circles, ellipses, hyperbolas, and parabolas. It provides the general equation for conic sections and the conditions to determine which type of conic section is represented based on the values of certain coefficients in the equation. It then gives the standard forms of the equations for each type of conic section and discusses some of their defining geometric properties.
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The document discusses ellipses and their key properties. It defines an ellipse as the set of points where the sum of the distances to two fixed foci is constant. Examples are given showing how to identify the center, vertices, covertices, and foci of ellipses given their equations. It also explains how to write the standard form of the equation of an ellipse and determines whether it has a horizontal or vertical major axis. Activities are provided for students to practice identifying parts of ellipses and writing their standard forms from equations.
Conic Sections
The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone.
Circles
Conic sections are shapes formed by the intersection of a plane and a double cone. A hyperbola occurs if the plane cuts through both cones. A parabola occurs if the plane is parallel to the edge of the cone. An ellipse occurs if the plane is not parallel or cutting through both cones. A circle is a special case of an ellipse where the plane is perpendicular to the altitude of the cone.
1-ELLIPSE.pptx
The Kaybiang Tunnel is the longest elliptical shaped tunnel in the Philippines, connecting two towns. An ellipse is the set of all points in a plane where the sum of the distances from two fixed points, called foci, is a constant. The Kaybiang Tunnel has an elliptical shape with its longest section piercing through a mountain.
ELLIPSE.pptx
This document provides information about ellipses. It defines an ellipse as the set of all points where the sum of the distances from two fixed points (foci) is constant. It identifies the major axis, minor axis, vertices, center, and focal axis of an ellipse. It explains that the eccentricity of an ellipse is between 0 and 1, unlike circles which have an eccentricity of 0 or parabolas which have an eccentricity of 1. The document provides an example of finding the foci, axes, latus rectum, eccentricity, and equation of directrices for a given ellipse. It encourages asking any questions and thanks the reader.
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
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Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
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1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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ellipse (An Introduction)
- 2. Introduction An ellipse is one of the conic sections; Its shape is a bounded curve which looks like a flattened circle. The orbits of the planets in our solar system around the sun happen to be elliptical in shape. Also, just like parabolas, ellipses have reflective properties that have been used in the construction of certain structures.
- 3. Definition and Equation of an Ellipse Consider the points F1(3,0) and F2(3,0), as shown in Figure 1.19. What is the sum of the distances of A(4,2.4) from F1 and from F2? How about the sum of the distances of B and C(0,4) from F1 and from F2?
- 4. REMEMBER THIS! An ellipse is the set of all points on the plane, the sum of whose distances from two fixed points is a constant.
- 5. Definition and Equation of an Ellipse Let F1 and F2 be two distinct points. The set of all points P, whose distances from F1 and from F2 add up to a certain constant, is called an ellipse. The pointsF1 and F2 are called the foci of the ellipse.
- 6. GRAPH OF AN ELLIPSE WITH STANDARD EQUATION Here the features of the graph of an ellipse with standard equation where a > b. Let 𝑐 = 𝑎2 − 𝑏2 . 𝑥2 𝑎2 + 𝑦2 𝑏2 = 1
- 7. GRAPH OF AN ELLIPSE WITH STANDARD EQUATION (1) Center: origin (0,0) (2) foci: F1(-c,0) and F2(c,0) (3) vertices: V1(-a,0) and V2(a,0) (4) covertices W1(0,-b) and W2(0,b)
- 8. ELLIPSE WITH STANDARD EQUATION Example 1. Give the coordinates of the foci, vertices, and covertices of the ellipse with equation 𝑥2 25 + 𝑦2 9 = 1. Sketch the graph, and include these points.
- 9. ELLIPSE WITH STANDARD EQUATION Example 2. Find the (standard) equation of the ellipse whose foci are F1(3,0) and F2(3,0), such that for any point on it, the sum of its distances from the foci is 10.
- 10. TRY THIS! 1. Give the coordinates of the foci, vertices, and covertices of the ellipse with equation 𝑥2 169 + 𝑦2 25 = 1. Sketch the graph, and include these points. 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 its distances from the foci is 20.