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For ellipses and hyperbolas a standard form has the x-axis as principal axis and the origin (0,0) as center. The vertices are (±a, 0) and the foci (±c, 0). Define b by the equations c 2 = a 2 − b 2 for an ellipse and c 2 = a 2 + b 2 for a hyperbola. For a circle, c = 0 so a 2 = b 2, with radius r = a = b.
Then for the ellipse case of AC > (B/2) 2, the ellipse is real if the sign of K equals the sign of (A + C) (that is, the sign of each of A and C), imaginary if they have opposite signs, and a degenerate point ellipse if K = 0. In the hyperbola case of AC < (B/2) 2, the hyperbola is degenerate if and only if K = 0.
Menaechmus likely discovered the conic sections, that is, the ellipse, the parabola, and the hyperbola, as a by-product of his search for the solution to the Delian problem. [3] Menaechmus knew that in a parabola y 2 = L x, where L is a constant called the latus rectum , although he was not aware of the fact that any equation in two unknowns ...
(The parabolas are orthogonal for an analogous reason to confocal ellipses and hyperbolas: parabolas have a reflective property.) Analogous to confocal ellipses and hyperbolas, the plane can be covered by an orthogonal net of parabolas, which can be used for a parabolic coordinate system .
F: focus of the red parabola and vertex of the blue parabola. In geometry, focal conics are a pair of curves consisting of [1] [2] either an ellipse and a hyperbola, where the hyperbola is contained in a plane, which is orthogonal to the plane containing the ellipse. The vertices of the hyperbola are the foci of the ellipse and its foci are the ...
Examples: The orthoptic of a parabola is its directrix (proof: see below),; The orthoptic of an ellipse + = is the director circle + = + (see below),; The orthoptic of a hyperbola =, > is the director circle + = (in case of a ≤ b there are no orthogonal tangents, see below),
A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ...
The ellipse, parabola, and hyperbola are viewed as conics in projective geometry, and each conic determines a relation of pole and polar between points and lines. Using these concepts, "two diameters are conjugate when each is the polar of the figurative point of the other." [5] Only one of the conjugate diameters of a hyperbola cuts the curve.
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