Parabolas

    In general, a parabola is a conic sections that can generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. In this topic we define a parabola as locus of points which are equidistant from a given point (the focus) and a given line (the directrix). We illustrate the definition with several examples including how to find an equation of a parabola given some geometric information. Conversely, we also show how to find the vertex, directrix, and focus given the equation of the parabola.  

Definition (Parabola)  A parabola is the set of points in a plane equidistant from a fixed point (the focus) and a fixed line (the directrix) that lie in the plane. The axis of the parabola is the line through the focus that is perpendicular to the directrix. The vertex of the parabola is the point on the axis halfway from to

Proposition (Parabola Centered at the Origin)  
    (i) An equation of a parabola centered at the origin with horizontal directrix and focus is If then the graph of the parabola opens upward and if then the graph of the parabola opens downward.
    (ii) An equation of a parabola centered at the origin with vertical directrix and focus is If then the graph of the parabola opens rightward and if then the graph of the parabola opens leftward.

Proposition (Parabola Equation)  
    (i) An equation of a parabola centered at with horizontal directrix and focus is If then the graph opens upward and if then the graph of the parabola opens downward.
    (ii) An equation of a parabola centered at with vertical directrix and focus is If then the graph opens rightward and if then the graph of the parabola opens leftward.

Example (Find the Vertex, Directrix, and Focus of the Parabola)  For each of the following equations of the parabola, find the vertex, directrix, and the focus:

(a)
    Solution. We have the form and so Therefore, the graph of the parabola opens rightward, the vertex is the directrix is the vertical line and the focus is
    
(b)
    Solution. We have the form and so with Therefore, the graph of the parabola opens leftward, the vertex is the directrix is the vertical line and the focus is

(c)
    Solution. To have the form we first complete the square:



Thus, our equation is So with Therefore, the graph of the parabola opens leftward, the vertex is the directrix is the vertical line and the focus is

Example (Find the Equation of the Parabola)  Find an equation of the parabola that satisfies the geometric conditions:

(a) The focus is and the directrix is
    Solution. The directrix is a horizontal line so the equation of the parabola has the form We need to determine Since the focus is and the vertex is the graph of the parabola is centered at the origin. Thus the equation is

(b) The focus is and the directrix is
    Solution. The directrix is a horizontal line so the equation of the parabola has the form Since the vertex is the halfway between the directrix and the focus we find that and the graph opens downward. Thus the graph of the parabola is centered at and so the equation of the parabola is

(c) The vertex is the axis is parallel to the -axis, and the parabola passes through the point
    Solution. The directrix is a vertical line so the equation of the parabola has the form Since the vertex is given we have, Thus, To find we use the given point to obtain and so Therefore, the equation is

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Parabolas
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/parabolas.html