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A circle is the set of all points in a plane that are equidistant from a fixed point.
A parabola is the set of all points in a plane equidistant from a given point and a given line.
opens up/down: y - k = (1 / 4p)(x - h)^2
opens left/right: x - h = (1 / 4p)(x - h)^2
p = distance from vertex to focus
An ellipse is the set of all point in a plane such that the sum of the distance to two fixed points is a constant.
horizontal: [(x - h)^2 / a^2] + [(y - k)^2 / b^2] = 1
Vertical: [(x - h)^2 / b^2] + [(y - k)^2 / a^2] = 1
a^2 - b^2 = c^2
A hyperbola is the set of all points in a plane such that the difference to two fixed points is a constant.
left/right: [(x - h)^2 / a^2] - [(y - k)^2 / b^2] = 1
Up/down: [(y - k)^2 / a^2] - [(x - h)^2 / b^2] = 1
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