After dividing the A2B2):That is the final equation for the central formula of the ellipse. The general equation would be derived by substituting in the original distance formulas the distance the ellipse is moved at either the domain or the range from the original C, in which the final general formula is located below (h being the domain moved, k being the range difference:(Gullberg, 560-561)Next, the derivation of the central and general formulas for the parabola will be discussed. The parabola is considered to be the locus of points equal distance from fixed point (the focus), and a line (the directrix). The distance from the focus to any point on the parabola will always be equal to the distance from the directrix to that parabola. The vertex as labeled below is considered to be on the axis of symmetry, or the midpoint of the line extending from the focus to the midpoint. See the diagram below for visual aid:The central equation is used to get a constant for which the certain points on the line can be determined and eventually leading to the capability of drawing a parabola. There is a quite simple formula that is commonly used, and it is: X2 = 4PY. Taking the distance formula and setting the distance of the central point equal to the distance to the directrix derives this formula. The directrix is Y = -P so when using the formula, the distance of the point on the parabola to the directrix is equal to Y + P. With this known, the formula below is derived when the two distances are set equal to each other:((X V 0)2 + (Y V P)2 = Y - (-P)Which can be simplified to:X2 = 4PYThe general equation would then be equal to the same formula, but in the initial distance formula one would subtract the new point, (U, V) from the X and Y, as illustrated below (after being simplified):(X V U)2 = 4P(Y V V)(Gullberg, 563)The final specific conic section that will be discussed is the hyperbola. The hyperbola is a locus of points for which the ...
