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The equation is for an ellipse, since both eigenvalues are positive. (Otherwise, if one were positive and the other negative, it would be a hyperbola.) The principal axes are the lines spanned by the eigenvectors. The minimum and maximum distances to the origin can be read off the equation in diagonal form.
For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in ...
An ellipse (red) obtained as the intersection of a cone with an inclined plane. Ellipse: notations Ellipses: examples with increasing eccentricity. In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.
The line with equation ax + by + c = 0 has slope -a/b, so any line perpendicular to it will have slope b/a (the negative reciprocal). Let (m, n) be the point of intersection of the line ax + by + c = 0 and the line perpendicular to it which passes through the point (x 0, y 0). The line through these two points is perpendicular to the original ...
Take P to be the origin. For a curve given by the equation F(x, y)=0, if the equation of the tangent line at R=(x 0, y 0) is written in the form + = then the vector (cos α, sin α) is parallel to the segment PX, and the length of PX, which is the distance from the tangent line to the origin, is p.
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.
where A xx, A xy, A yy, B x, B y, and C are the constants defining the equation. For such a conic section, the polar line to a given pole point (ξ, η) is defined by the equation + + = where D, E and F are likewise constants that depend on the pole coordinates (ξ, η)
The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The distance between the vertex ...