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The 5-, 4- and 3- point degenerations of Pascal's theorem are properties of a conic dealing with at ... there is no parabola with directrix parallel to the x axis, ...
The ellipse thus generated has its second focus at the center of the directrix circle, and the ellipse lies entirely within the circle. For the parabola, the center of the directrix moves to the point at infinity (see Projective geometry). The directrix "circle" becomes a curve with zero curvature, indistinguishable from a straight line.
For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix the line with equation x = −a. In standard form the parabola will always pass through the origin. For a rectangular or equilateral hyperbola, one whose asymptotes are perpendicular, there is an alternative standard form in which the ...
It has been proved that the Kiepert hyperbola is the hyperbola passing through the vertices, the centroid and the orthocenter of the reference triangle and the Kiepert parabola is the parabola inscribed in the reference triangle having the Euler line as directrix and the triangle center X 110 as focus. [1]
A family of conic sections of varying eccentricity share a focus point and directrix line, including an ellipse (red, e = 1/2), a parabola (green, e = 1), and a hyperbola (blue, e = 2). The conic of eccentricity 0 in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity ∞ is an infinitesimally separated ...
where (h, k) is the center of the ellipse in Cartesian coordinates, in which an arbitrary point is given by (x, y). The semi-major axis is the mean value of the maximum and minimum distances r max {\displaystyle r_{\text{max}}} and r min {\displaystyle r_{\text{min}}} of the ellipse from a focus — that is, of the distances from a focus to the ...
A: vertex of the red parabola and focus of the blue parabola 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 ...
Neither Dandelin nor Quetelet used the Dandelin spheres to prove the focus-directrix property. The first to do so may have been Pierce Morton in 1829, [8] or perhaps Hugh Hamilton who remarked (in 1758) that a sphere touches the cone at a circle which defines a plane whose intersection with the plane of the conic section is a directrix.