e directrix are the same, the eccentricity is therefore equal to one. In the ellipse, it is less than one because the A is always greater than the C. In the hyperbola the eccentricity is always greater than one because the opposite of the ellipse occurs. In the circle, the eccentricity is always equal to zero. The names for each of the conic sections came directly from their eccentricities. For example, the circle gets its name from the Greek word kirkos, meaning a ring. The ellipse gets its name from the Greek word elleipein, meaning to deduct, the parabola from the Greek word paraballein, and the hyperbola from the Greek word hyperballein, meaning to exceed, or be in excess. In using these noted eccentricities, doing some substitutions, rotating of the axes, and a lot of simplifying, you come up with a shorter version of the vertex equation. But to make it work for every conic section, you must add the V (1 V E2) X2 to the Y2 = 2PX. Therefore you will result in the final vertex equation of: Y2 = 2PX V (1 V E2) X2There are many ways in which conic sections apply to our environment and the rest of our surroundings and the things in them. Foremost, the ellipse has been studied as a part of the universal geometric code since in 1609 when a German astronomer by the name of Johannes Kepler proved with astronomical data that the patterns of travel by an object, a planet, in space, could be described by using a geometric curve. This curve was then known as an ellipse. The studies that were then culminated led to the creation of three separate laws that relate conic sections to planetary motion. These laws are today known as Keplers Laws. The first law of planetary motion pretty much stated that planets move in an elliptical pattern around their focal object, in this case, the sun. The second law of planetary motion explains that there is a steady rate of time and distance at which the distance of the object from the focus chan...
