Conic Sections
Conic sections correspond to slices of cones:
Parabolas are the loci of points equidistant from points and lines. The parabola of (0, f) and y = -f corresponds to x2 + (y - f)2 = (y + f)2 or y = x2 / (4f).
Circles are the loci of points equidistant from points. The circle of the origin and r corresponds to x2 + y2 = r2.
Ellipses are the loci of points where the sums of distances from points are equal. The ellipse of (0, -f), (0, f) and 2a corresponds to √(x + f)2+ y2 + √(x - f)2 + y2 = 2a or x2 / a2 + y2 / b2 = 1 such that b2 = a2 - f2.
Hyperbolas are the loci of points where the differences of distances from points are equal. The hyperbola of (0, -f), (0, f) and 2a corresponds to √(x + f)2+ y2 - √(x - f)2 + y2 = 2a or x2 / a2 - y2 / b2 = 1 such that b2 = f2 - a2.
Parabolas are the loci of points equidistant from points and lines. The parabola of (0, f) and y = -f corresponds to x2 + (y - f)2 = (y + f)2 or y = x2 / (4f).
Circles are the loci of points equidistant from points. The circle of the origin and r corresponds to x2 + y2 = r2.
Ellipses are the loci of points where the sums of distances from points are equal. The ellipse of (0, -f), (0, f) and 2a corresponds to √(x + f)2+ y2 + √(x - f)2 + y2 = 2a or x2 / a2 + y2 / b2 = 1 such that b2 = a2 - f2.
Hyperbolas are the loci of points where the differences of distances from points are equal. The hyperbola of (0, -f), (0, f) and 2a corresponds to √(x + f)2+ y2 - √(x - f)2 + y2 = 2a or x2 / a2 - y2 / b2 = 1 such that b2 = f2 - a2.
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