page_title: Ellipse
section_title: Theorem of Apollonios on conjugate diameters
hierarchical_section_title: Ellipse / Conjugate diameters / Theorem of Apollonios on conjugate diameters
context_page_description: 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. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e, a number ranging from e = 0 to e = 1.
  Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is:
  Assuming a ≥ b, the foci are for. The standard parametric equation is:
  Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane. Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a cylinder is also an ellipse.
  An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity:
  .
context_section_description: For an ellipse with semi-axes  the following is true:
  Let  and   be halves of two conjugate diameters (see diagram) then
  ,
  the triangle formed by  has the constant area 
  the parallelogram of tangents adjacent to the given conjugate diameters has the 
  Proof
  Let the ellipse be in the canonical form with parametric equation
  .
  The two points  are on conjugate diameters (see previous section). From trigonometric formulae one obtains  and
  The area of the triangle generated by  is
  and from the diagram it can be seen that the area of the parallelogram is 8 times that of . Hence
caption: Ellipse: theorem of Apollonios on conjugate diameters