1. Definition of ellipse: An ellipse is the locus of a point on a plane whose sum of the distances from one point to two fixed points F 1 and F2 is equal to a constant (greater than |F 1F2|). These two fixed points are called focal points and the constant is called focal length.
2. Standard equation of ellipse: There are two forms of standard equation of ellipse, namely, general equation of ellipse and parametric equation of ellipse. The general equation is (x-h)_/a_+(y-k)_/b_= 1, where (h, k) is the center of the ellipse and a and b are the radii of the major axis of the ellipse. The parameter equation is x=h+a*cosθ, y=k+b*sinθ, where θ is the parameter.
3. Properties of ellipse: Ellipse has many important properties, such as eccentricity, focus, directrix, etc. Eccentricity is the ratio of the distance from any point on the ellipse to the focus and the distance from the directrix, which reflects the flatness of the ellipse. The focal point is two special points on the ellipse, and their distance from the center of the ellipse is equal to half the focal length. The directrix is a straight line perpendicular to the line where the focus is located and at the same distance from the focus.
4. Tangent of the ellipse: Any point on the ellipse can be tangent to two straight lines, one parallel to the X axis and the other parallel to the Y axis. The slope of these two tangents is the derivative of the ellipse at this point.
5. Area of ellipse: The area of ellipse can be calculated by integration, and the formula is S=πab, where A and B are the major axis and minor axis of ellipse respectively.
6. Application of Ellipse: Ellipse is widely used in many fields, such as astronomy, physics, engineering and so on. For example, the orbits of planets can usually be approximated by ellipses, so ellipses play an important role in astronomy.