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Conics, Parabolas, Ellipses and Hyperbolas

 Topic Review on "Title": Definition of a parabola: A parabola is a set of all points in a plane that are equidistant from a given fixed point (the Focus) and a given straight line (the Directrix). Different cases of parabolas: 1) With the vertex at the origin, the parabola opens in the positive x direction and has the equation where vertex=(0,0) and focus is the point (p,0). 2) With the vertex at the origin, the parabola opens in the negative x direction and has the equation where vertex=(0,0) and focus is the point (p,0). 3) With the vertex at the origin, the parabola opens in the positive y direction and has the equation where vertex=(0,0) and focus is the point (0,p). 4) With the vertex at the origin, the parabola opens in the negative y direction and has the equation  where vertex=(0,0) and focus is the point (0,p). Definition of an ellipse: An ellipse is a set of all points in a plane, whose distances from two fixed points (the foci) is a positive constant. Different cases of ellipses: 1) The vertex is at the origin and the foci and the major axis are on the x-axis with the center at the origin and has the equation of the form where the foci and the major axis are on the x-axis, the length of the major axis is 2a, the minor  axis is on the y-axis, the length of minor axis equals to 2b and the center of the origin is at the origin (0,0). 2) The vertex is at the origin and the foci and the major axis are on the y-axis with the center at the origin and has the equation of the form where the foci and the major axis are on the y-axis, the length of the major axis=2a, the minor axis is on the x-axis, length of the minor axis=2b and the center is at the origin (0,0). Definition of a hyperbola: A hyperbola is a set of all points in a plane, the difference of whose distances from two fixed points (the foci) is a positive constant. Different cases of hyperbolas: 1) The center is at the origin and the foci are on the x-axis and conjugate axis is the y-axis and has the equation of the form where the foci and the vertices are on the x-axis, the distance between the foci=2a, the conjugate axis is on the y-axis and the center is at the origin (0,0). 2) The center is at the origin and the foci are on the y-axis and conjugate axis is the x-axis and has the equation of the form  where the foci and the vertices are on the y-axis, the distance between the foci=2a, the conjugate axis is on the x-axis and the center is at the origin. Asymptotic Equations: The equations of the asymptotes to the hyperbola  are as follows and .

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 "Title" Tutorial Summary : This tutorial introduces the different characteristics of the conic sections and the ways they are applied to particular problems. Graphs are shown to exhibit the history of conic sections and their use in several branches of mathematics. Key conic sections such as a parabola and a hyperbola are introduced as they are related to Calculus based problems. This tutorial discusses the ideas of hyperbolas and ellipses as they are related to different cases of the conic sections. Different scenarios where ellipses, hyperbolas change direction as far as their graphs are discussed in the sections of the tutorial. The foci and vertices of hyperbolas and ellipses are important to the motivation of the graphs of the different conic sections.

 Tutorial Features: Specific Tutorial Features: Step by step examples of the different aspects of the conic sections. “Completion of square” trick is used to bring the equations of the conic sections in standard form. Series Features: Concept map showing inter-connections of new concepts in this tutorial and those previously introduced. Definition slides introduce terms as they are needed. Visual representation of concepts Animated examples—worked out step by step A concise summary is given at the conclusion of the tutorial.

 "Title" Topic List: `Introduction to Conic SectionsA Brief History of the Conic SectionsSome applicationsParabola Definition of a parabola Different types of equations of a parabolaEllipse Definition of an ellipse Relationship between foci and axes Different types of equations of an ellipseHyperbola Definition of a hyperbola Relationship between foci and vertices Different types of equations of a hyperbola Asymptotes of a hyperbola`

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