Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 6Because P and Q both lie on the ellipse, the sum of the distances from each of them to thefoci must be equal. Thus, we obtainfrom which it follows thator equivalently2 √ b 2 +c 2 = (a−c)+(a+c)a = √ b 2 +c 2 (1.5)c = √ a 2 −b 2 (1.6)From (1.5), the distance from the focus to an end of the minor axis is a, which implied thatall points on the ellipse the sum of the distance to the foci is 2a.The equation of an ellipse is simplest if the center of the ellipse is at the origin and thefoci are on the x-axis or y-axis. The two possible such orientations are shown in Figurebelow. These are called the standard positions of an ellipse, and the result equationsare called the standard equations of an ellipse.•−a(−c,0)Orientationyb•(c,0)axStandard Equationx 2a 2 + y2b 2 = 1−bya•(0,c)−bbxx 2b 2 + y2a 2 = 1•(0,−c)−aTo illustrate how the equations in the above Figure are obtained, we will derive theequation for the ellipse with the foci on the x-axis.