Parabola jen


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Parabola jen

  1. 1. PARABOLA<br />In mathematics, the parabola (plural parabolae or parabolas, pronounced /pəˈræbələ/, from the Greek παραβολή) is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Given a point (the focus) and a corresponding line (the directrix) on the plane, the locus of points in that plane that are equidistant from them is a parabola. The line perpendicular to the directrix and passing through focus (that is the line that splits the parabola through the middle)is called the " axis of symmetry" . The point on the axis of symmetry that intersects the parabola is called the " vertex" . The vertex is the point where the parabola changes directions. Parabolas can open up, down, left, or right.<br />The name " parabola" is derived from a New Latin term that means something similar to " compare" or " balance" .<br />Conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. Each of those lines is called a generatrix of the surface.<br />Foci (pronounced /ˈfoʊsaɪ/, singular focus) are a pair of special points used in describing conic sections. The four types of conic sections are the circle, ellipse, parabola, and hyperbola.<br />The circle has eccentricity 0, and the directrix is a line at infinity. The focus-directrix property is thus true of the circle, but it is also true of every other point on the plane.<br />Conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. It can be defined as the locus of points whose distances are in a fixed ratio to some point, called a focus, and some line, called a DIRECTRIX.<br />The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.<br />Locus (Latin for " place" , plural loci) is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point.<br />A locus may alternatively be described as the path through which a point moves to fulfill a given condition or conditions. So, for example, a circle may also be defined as the locus of a point moving so as to remain at a given distance from a fixed point.<br />History of PArabola<br />The earliest known work on conic sections was by Menaechmus in the fourth century B.C.. He discovered a way to solve the problem of doubling the cube using parabolae. (The solution, however, does not meet the requirements imposed by compass and straightedge construction.) The name " parabola" is due to Apollonius, who discovered many properties of conic sections. The focus–directrix property of the parabola and other conics is due to Pappus.<br />Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.<br />The idea of using a paraboloid in a reflecting telescope is due to James Gregory in 1663 and the first to be constructed was by Isaac Newton in 1668. The same principle is used in satellite dishes and radar receivers.<br />1454150467360Let the directrix be the line x = −p and let the focus be the point (p, 0). If (x, y) is a point on the parabola then, by Pappus' definition of a parabola, it is the same distance from the directrix as the focus; in other words: <br />163957050800<br />1454150417830By translation, the general equation of a parabola with a horizontal axis is<br />1209040322580and interchanging the roles of x and y gives the corresponding equation of a parabola with a vertical axis as<br />1209675198755The last equation can be rewritten<br />so the graph of any function which is a polynomial of degree 2 in x is a parabola with a vertical axis.<br />624840295275More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation of the form<br />89090553975<br />where all of the coefficients are real, neither A nor B is zero and more than one solution exists, defining a pair of points (x, y) on the parabola. That the equation is irreducible means it does not factor as a product of two not necessarily distinct linear equations.<br />EQUATIONS<br />(with vertex (h, k) and distance p between vertex and focus - note that if the vertex is below the focus, or equivalently above the directrix, p is positive, otherwise p is negative; similarly with horizontal axis of symmetry p is positive if vertex is to the left of the focus, or equivalently to the right of the directrix)<br />Cartesian<br />Vertical axis of symmetry<br />476250-4371<br />.<br />Horizontal axis of symmetry<br />.<br />General parabola<br />228600024765The general form for a parabola is<br />474345234950This result is derived from the general conic equation given above:<br />1580515201295and the fact that, for a parabola,<br />Equation for a general parabola with a focus point F(u, v), and a directrix in the form<br />476250178095is<br />PARABOLA<br />1898044338352<br />A parabola (plural " parabolas" ; Gray 1997, p. 45) is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus). The focal parameter (i.e., the distance between the directrix and focus) is therefore given by , where is the distance from the vertex to the directrix or focus. The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a paraboloid. <br />For a parabola opening to the right with vertex at (0, 0), the equation in Cartesian coordinates is <br />(1) (2) <br />(3) (4) <br />The quantity is known as the latus rectum. If the vertex is at instead of (0, 0), the equation of the parabola is <br />(5) <br />If the parabola instead opens upwards, its equation is <br />(6) <br />1465078-4593<br />Three points uniquely determine one parabola with directrix parallel to the -axis and one with directrix parallel to the -axis. If these parabolas pass through the three points , , and , they are given by equations <br />(7) <br />and <br />(8) <br />In polar coordinates, the equation of a parabola with parameter and center (0, 0) is given by <br />(9) <br />(left figure). The equivalence with the Cartesian form can be seen by setting up a coordinate system and plugging in and to obtain <br />(10) <br />Expanding and collecting terms, <br />(11) <br />so solving for gives (◇). A set of confocal parabolas is shown in the figure on the right. <br />In pedal coordinates with the pedal point at the focus, the equation is <br />(12) <br />The parabola can be written parametrically as <br />(13) (14) <br />or <br />(15) (16) <br />Given an arbitrary point located " outside" a parabola, the tangent or tangents to the parabola through can be constructed by drawing the circle having as a diameter, where is the focus. Then locate the points and at which the circle cuts the vertical tangent through . The points and (which can collapse to a single point in the degenerate case) are then the points of tangency of the lines and and the parabola (Wells 1991). <br />The curvature, arc length, and tangential angle are <br />(17) (18) (19) <br />The tangent vector of the parabola is <br />(20) (21) <br />The parabola is defined as the locus of a point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix).<br />In the following graph, <br />The focus of the parabola is at (0, p).<br />The directrix is the line y = -p. <br />The focal distance is |p| (Distance from the origin to the focus, and from the origin to the directrix. We take absolute value because distance is positive.) <br />The point (x, y) represents any point on the curve.<br />The distance d from any point (x, y) to the focus (0, p) is the same as the distance from (x, y) to the directrix. <br />[The word locus means the set of points satisfying a given condition.<br />The Formula for a Parabola - Vertical Axis <br />Adding to our diagram from above, we see that the distance d = y + p. <br />Now, using the distance formula on the general points (0, p) and (x, y), and equating it to our value d = y + p, we have<br />Squaring both sides gives:<br />(x − 0)2 + (y − p)2 = (y + p)2<br />Simplifying gives us the formula for a parabola:<br />x2 = 4py<br />In more familiar form, with " y = " on the left, we can write this as:<br />where p is the focal distance of the parabola. <br />Example:<br />The focal length is found by equating the general expression for y <br />and our particular example:<br />So we have: <br />This gives p = 0.5.<br />So the focus will be at (0, 0.5) and the directrix is the line y = -0.5.<br />Our curve is as follows: <br />Note: Even though the sides look as though they become straight as x increases, in fact they do not. The sides of a parabola just get steeper and steeper (but are never vertical, either). <br />