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Parabolas are also one of the four curves (circle, parabola, ellipse, and hyperbola) that are called conic sections simply because they were originally obtained by slicing a right circular cone with a plane. If the plane is parallel to one element of the cone, the resulting intersection is a parabola.
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A more direct definition of a parabola is as follows: a parabola is the set of all points in a plane that are equidistant from a fixed point and a fixed line. The fixed point is called the focus and the fixed line is called the directrix. The midpoint of the perpendicular segment from the focus to the directrix is the vertex of the parabola. The line that passes through the vertex and the focus is the axis of symmetry.
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If we choose (a,0) as the focus and the line x = b as the directrix, any point on the parabola, (x, y), needs to satisfy the following equation:
x - b = [(x-a)2 + y2]1/2
because the distance from the directrix is equal to the distance from the focus.
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Simplifying the equation, we have:
x = [1/(2a-2b)] y2 + (a+b)/2
or
x = c y2 + d
in which c = 1/(2a-2b) and d = (a+b)/2.
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