qual occurs. This is shown after occurring below:B2X2 V A2Y2 = A2B2 Then, after dividing both of the sides by A2B2, you receive the final formula:By modifying the formula so that the center of the equation can be moved to elsewhere on the graph, and not restricted to just the origin, you can name the new central point (H, K) and come up with a new formula that would yield the general equation. This is done by placing the H and the V in places in the original distance formula like the central equation, then working the problem out again by squaring the sides and simplifying many times. An example of what one would start with when deriving the general equation would be:((X V(H V C))2 + (Y V K)2) + ((X V (H + C)2 + (Y V K)2) = 2AAfter some of the reducing we have known and grown to love, you get this:(Gullberg, 566-567)That is the final general equation for the hyperbola not centered at the origin, or the point on an ordinary graph labeled (0, 0). There is a formula that is used to come up with the eccentricity of any type of conic section, with just a single formula. This equation is known as the vertex equation. The vertex equation is:Y2 = 2PX V (1 V E2) X2The letters all are the same as mentioned earlier, but the E stands for eccentricity. The eccentricity is derived from the ratio of the distance of the focus to the point on the curve, to the distance of the directrix to the same point on the curve. Examples below:The formula for eccentricity in this graph is:The eccentricities for each of the pairs of ratios remains constant as noted because the ellipse is still the same. There is also a formula for the eccentricity of a hyperbola. The formula for eccentricity of this conic section is C over A. It is proven through distance formulas and substitutions. Also shown, is that the directrixes have the equations of positive and negative A over E. With the parabola, the distance from the point on a curve to the focus and to th...
