the radius. This is known as the central equation, X2 + Y2 = r2. The way this formula came about was by expressing the radius in terms of the distance formula: r = (X-0)2 + (Y-0)2 , with the right half of the equation under the square root. In taking both sides and squaring them, you result in r2= (X-0) 2+ (Y-0) 2. To simplify this equation, when the center is at the origin it is not necessary to subtract the x and y coordinate of the center of the circle. (Gullberg, 559) For example, if the center of the circle was at (3, -5), then the equation would be r = (X-3)2 + (Y+5)2. That is how you would use the equation is the center of the circle was not located at the origin.Next, the ellipse and the derivation of its central and general formulas will be discussed. The ellipse is considered the locus of points for which the sum of their distances from two fixed points remain the same, or constant. In the diagram below, F1 and F2 are Foci. For purposes of space, the diagram below will also include the material on the central equation for an ellipse. Lets say that point P, or (X, Y), is on the perimeter of the ellipse. Then the distance from F-1 to P + the distance from F2 to P added together equals the constant, which also equals the length of the major axis (The farthest end of the ellipse to the other end). The major axis will be called 2A in terms of there being two parts extending from the center of the ellipse, which would also be the same as saying that 2A is equal to the sum of the distances from the foci to point P. From there, one could say that the length of the minor axis, or the line that is labeled Y on the diagram, is B2, and the distance from the center from the center of the ellipse to the focus is C2. Now the A is equal to the focus to the interception of the minor axis on the circle because if that was to be doubled, it would equal the same as the sum of the distances of any point on the perimeter since that n...
