Plane and solid analytic geometry; an elementary textbook . ellipse Fig. 64. b2x2 + a2y2 = a2b2,its coordinates must satisfy this equation. Hence Vx^^-T^ a? Oh. IX, § 70] CONIC SECTIONS 119 Substituting this value of y^ in the expression for thedistance, Ave have FP1 = yjx2 - 2 aexx + a2e2 + b2 - h\ x2. a2 — b2 Noting that — = e2, and a2e2 + b2 = a2, a2 this reduces to FPX = Va2 - 2 aex1 + e2x2 = ±(a- exj. The distance, FPV of the point from the left-handfocus may be found in the same way except that the coor-dinates of F are (— ae, 0). Hence FPl = ± (a + ex{). Let the student show that, thoug
Plane and solid analytic geometry; an elementary textbook . ellipse Fig. 64. b2x2 + a2y2 = a2b2,its coordinates must satisfy this equation. Hence Vx^^-T^ a? Oh. IX, § 70] CONIC SECTIONS 119 Substituting this value of y^ in the expression for thedistance, Ave have FP1 = yjx2 - 2 aexx + a2e2 + b2 - h\ x2. a2 — b2 Noting that — = e2, and a2e2 + b2 = a2, a2 this reduces to FPX = Va2 - 2 aex1 + e2x2 = ±(a- exj. The distance, FPV of the point from the left-handfocus may be found in the same way except that the coor-dinates of F are (— ae, 0). Hence FPl = ± (a + ex{). Let the student show that, though the work in the caseof the hyperbola will be slightly different, the results willbe the same. Since it is only the length of the focal radii that weseek, it will be neces-sary to determine ineach conic which signshould be used beforethe parenthesis, so thatit may express a posi-tive distance. In theellipse a is always greater than exv and the positive sign must therefore beused in both cases. Hence, in the ellipse, FPX = a- exuand FP1 = a + exu. [45] 120 ANALYTIC GEOMETRY [Ch. IX, § 71 It will be necessary to consider the two branches of thehyperbola separately. For the right-hand branch exx isalways positive and greater than a; and the two dis-tances are FP-. = exi - a, [46, »]and FP1 = exi + a. For the left-hand branch ex1 is negative and greater inabsolute value than a; and the two distances are FP, = — exi + a, [46, .5]and FrP1 = — exx - a. From these results we see that the sum of the two focalradii of any point of an ellipse is 2 a. While in the hyper-bola the difference of the tiro focal radii is 2 a. Theellipse might therefore be defined as the locus of points,the sum of whose distances from two fixed points is con-stant ; and the hyperbola as the locus of points, thedifference of whose distances from two fixed points isconstant. Let the student obtain the equations of the ellipse andhyperbola in their ordinary forms from these definitions. 71. Mechanic
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