Parabola

The parabola (from the generated by the intersection of a right circular Conical Surface and a Plane parallel to a generating straight line of that surface. A parabola can also be defined as Locus of Point s which are Equi Distant from a given point (the ''' Focus ''') and a given line (the '''directrix'''). A particular case arises when the plane is tangent to the conical surface. In that case the intersection is a Degenerate parabola consisting of a Straight Line . DEFINITIONS AND OVERVIEW Analytic geometry equations In Cartesian Coordinates , a parabola with an axis parallel to the ''y'' axis with vertex (''h'', ''k''), focus (''h'', ''k'' + ''p''), and directrix ''y'' = ''k'' - ''p'', with ''p'' being the distance from the vertex to the focus, has the equation : $(x - h)^2 = 4p(y - k) \,$ or, alternatively : $(y - k) = rac{1}{4p}(x-h)^2 \,$ More generally, a parabola is a curve in the Cartesian Plane defined by an Irreducible equation of the form : $A x^2 + B xy + C y^2 + D x + E y + F = 0$ such that $B^2 = 4 AC$ , where all of the coefficients are real, where A and/or C is non-zero, and where more than one solution, defining a pair of points (x, y) on the parabola, exists. That the equation is Irreducible means it does not factor as a product of two not necessarily distinct linear factors. Other geometric definitions A parabola may also be characterized as a conic section with an Eccentricity of 1. As a consequence of this, all parabolas are Similar . A parabola can also be obtained as the Limit of a sequence of Ellipse s where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction. In this sense, a parabola may be considered an ellipse that has one focus at Infinity . The parabola is an Inverse Transform of a Cardioid . A parabola has a single axis of reflective Symmetry , which passes through its focus and is perpendicular to its directrix. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a Paraboloid of revolution. The parabola is found in numerous situations in the physical world (see below). Equations (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 iff vertex is to the left of the focus, or equivalently to the right of the directrix) Cartesian =Vertical axis of symmetry : $(x - h)^2 = 4p(y - k) \,$ : $y = a(x-h)^2 + k \,$ : $y = ax^2 + bx + c \,$ :: $\mbox{where }a = rac{1}{4p}; \ \ b = rac{-h}{2p}; \ \ c = rac{h^2}{4p} + k; \ \$ :: $h = rac{-b}{2a}; \ \ k = rac{4ac - b^2}{4a}$ . : $x(t) = 2pt + h; \ \ y(t) = pt^2 + k \,$ =Horizontal axis of symmetry : $(y - k)^2 = 4p(x - h) \,$ : $x = a(y - k)^2 + h \,$ : $x = ay^2 + by + c \,$ :: $\mbox{where }a = rac{1}{4p}; \ \ b = rac{-k}{2p}; \ \ c = rac{k^2}{4p} + h; \ \$ :: $h = rac{4ac - b^2}{4a}; \ \ k = rac{-b}{2a}$ . : $x(t) = pt^2 + h; \ \ y(t) = 2pt + k \,$ ''' Semi-rectum and polar coordinates In Polar Coordinates , a parabola with the focus at the origin and the top on the negative ''x''-axis, is given by the equation : $r (1 - \cos heta) = l \,$ where ''l'' is the '' Semi- Latus Rectum '': the distance from the focus to the parabola itself, measured along a line perpendicular to the axis. Note that this is twice the distance from the focus to the apex of the parabola or the perpendicular distance from the focus to the latus rectum. Gauss-mapped form A Gauss-mapped form: $( an^2\phi,2 an\phi)$ has normal $(\cos\phi,\sin\phi)$ . SEE ALSO Paraboloid Ellipse Hyperbola Conic Section DERIVATION OF THE FOCUS Given a parabola parallel to the ''y''-axis with vertex (0,0) and with equation : $y = a x^2, \qquad \qquad \qquad (1)$ then there is a point (0,''f'') — the focus — such that any point ''P'' on the parabola will be equidistant from both the focus and a line perpendicular to the axis of symmetry of the parabola (the ''linea directrix''), in this case parallel to the ''x'' axis. Since the vertex is one of the possible points P, it follows that the linea directrix passes through the point (0,-''f''). So for any point ''P=(x,y)'', it will be equidistant from (0,''f'') and (''x'',-''f''). It is desired to find the value of ''f'' which has this property. Let ''F'' denote the focus, and let ''Q'' denote the point at (''x'',-''f''). Line ''FP'' has the same length as line ''QP''.
	:<math> \ QP \	y + f </math>
	:<math> \ FP \	\ QP \ </math>

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Parabola