Elements of analytical geometry and the differential and integral calculus . e curve, confined by twofixed points, and the sum of the distances from any point in thecurve to the fixed points, is constant. 2.—The two fixed points are called the foci. 3.—The center is midway in a straight line between the twofoci. 4.—A diameter is a straight line passing through the center. 6.—The major axis is a diameter passing through the foci. 8.—The minor axis is at right angles to the major axis, passingthrough the center. 9.—The distance between the center and either/ocms, is calledthe eccentricity when t
Elements of analytical geometry and the differential and integral calculus . e curve, confined by twofixed points, and the sum of the distances from any point in thecurve to the fixed points, is constant. 2.—The two fixed points are called the foci. 3.—The center is midway in a straight line between the twofoci. 4.—A diameter is a straight line passing through the center. 6.—The major axis is a diameter passing through the foci. 8.—The minor axis is at right angles to the major axis, passingthrough the center. 9.—The distance between the center and either/ocms, is calledthe eccentricity when the semimajor axis is unity. 10. —The parameter of an ellipse is the double ordinate passingthrough one of the foci. PROPOSITIOIf I. To find the equation of the curve, the origin of the co-<yrdinaiesbeinff in the center, the major axis being given, also the distanceof the foci from the center. The curve in the margin repre-sents an ellipse. Put CF=c, CA= any point, as F, and let fall theperpendicular Ft. By our conventional notation, put Ci=x, tF= THE ELLIPSE. 43 As FP-\-PF=<2,A, we may put FP=A+z, and PF=A— the two right angled triangles FPt, FPt, give us{c+xY+y^^={A^zY (1) {c—xY+y^-={A—zy (2) For the points in the curve which cause t to fall between c andF, we would have {x-cY+y-=.{A-zY (3) But when expanded, there is no difference between (2) and(3), and by giving proper values and signs to x and y, equations(1) and (2) will respond to any point in the curve as well as tothe point P. Substracting (2) from (1), and dividing by 4, we find cx=Azy or 2= (4) A This last equation shows that FP, the radius vector, varies as the abscissa x. Add (1) and (2), and divide the sum by 2, and we have Substituting the value of z^ from (4), and clearing of frac-tions, we have c^A^+A-x^-^A^y^z^A^+c^xKOr Ay^-\-{A^—c^)x^=A-(A^—c^). (5) Now conceive the point P to move along describing the curve,and when it comes to the point B, so that D C makes a ri
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