Ellipses in Standard Form Worksheets
What Is Meant By Ellipses in the Standard Form?
An ellipse is the set of points in a plane whose distances from two fixed points, called foci, have a sum that is equal to a positive constant. In other words, if points F1F1 and F2F2 are the foci (plural of focus) and dd is some given positive constant then (x, y) (x, y) is a point on the ellipse if d= d1+d2 as pictured below:
In addition, an ellipse can be formed by the intersection of a cone with an oblique plane that is not parallel to the side of the cone and does not intersect the base of the cone. Points on this oval shape where the distance between them is at a maximum are called vertices and define the major axis. The center of an ellipse is the midpoint between the vertices. The minor axis is the line segment through the center of an ellipse defined by two points on the ellipse where the distance between them is at a minimum. The endpoints of the minor axis are called co-vertices.
The Specifics on Ellipses
An ellipse is defined as the set of all the points arranged in a plane such that the distance between points remains constant. Each fixed point is known as the foci of the ellipse. Ellipse has two axes of symmetry. The shorter axis is known as the minor axis, while the longer axis is known as the major axis. The endpoint of major axis is known as the vertex of the ellipse, while the endpoint of the minor axis is known as the co-vertex of the ellipse. The midpoint of the minor and major axes is the center of an ellipse. Both major and minor axes are perpendicular to the center, with foci lying on the major axis. The sum of the distances from foci to any point on the ellipse is greater than the distance between the two foci. The standard form of the equation of an ellipse with major axis parallel to x-axis and center at (0,0). x2 / a2 + y2 / b2 = 1 where a > b: The coordinates of the vertices are (± a, 0) The coordinates of co-vertices are (0, ±b) Length of major axis is 2a The length of the minor axis is 2b The coordinates of the foci are (±c, 0) The standard form of the equation of the ellipse with major axis parallel to y-axis and center at (0,0) x2 / b2 + y2 / a2 = 1 where a > b: The coordinates of the vertices are (0, ± a) The coordinates of co-vertices are (±b, 0) Length of major axis is 2a The length of the minor axis is 2b The coordinates of the foci are (0, ±c)
Carl Friedrich Gauss
"I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect...geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics."