COR. 1.-The square of a perpendicular, upon any diameter, from a point in the curve, is equal to the rectangle under the parameter of the axis, and the absciss corresponding to the ordinate from the same point. For it was proved that GK2=4CV • MN. COR. 2.-If there be two diameters, and from the vertex of each a semi-ordinate be applied to the other, the abscisses will be equal. For, let DK, and EH, produced (figure to proposition VI.) be the diameters; then GH is an ordinate to DK, and one from M applied to EH will be the other. Let the perpendicular GK upon DK be called P; and that from M upon EH, P'; the abscissa MN, A; and that of EH, A'. Then (Cor. 1) 4A CV P2, and 4A' · CV = P/2; but since GN NH, the perpendiculars P and P' are equal (Pl. Ge. I. 26); hence P2P2; and therefore 4A CV4A' CV, or A = A'. COR. 3.-The square of that part of a tangent, between the point of contact and any diameter, is equal to the rectangle under the external segment of that diameter, and the parameter of the diameter which passes through the point of contact. For RM=GN, and RG=MN (figure to proposition VI.) COR. 4. The squares of ordinates, or semi-ordinates, to any diameter, are to one another as their corresponding abscisses; and the squares of perpendiculars, from the same points, are in the same ratio. = = Let O, O', be two semi-ordinates to the same diameter; A, A', the corresponding abscisses, and P the parameter of the diameter. Then 02 PA, and 022 = = PA'; therefore 02:02 P· A: P· A'A: A' (Pl. Ge. VI. 1). COR. 5.-If the squares of parallel lines, drawn from certain points, to meet a line given in position, be to one another as the parts they cut off towards one extremity, these points will be in the curve of a parabola, which has the given line for a diameter. For if the parallel lines be considered as ordinates, and the segments cut off from the given line as abscissæ, it follows (Cor. 4) that the given line is a diameter to a parabola passing through the extremities of the ordinates. PROPOSITION VIII. A subtangent, upon any diameter, is bisected in the vertex of that diameter. Let the tangent GM meet any diameter VN in M, and let GN be an ordinate to it, from the point of contact, the subtangent MN is bisected in V (figure to proposition VII.) For, let the diameter GL, and its semi-ordinate VL, be drawn, then is the absciss GL = VN (I. 7, Cor. 2); but, since LM is a parallelogram (I. 6, Cor. 1), GL = MV; therefore MV = VN. COR. 1.-That part of the axis, between the focus and any tangent, is of the parameter of the diameter passing through the point of contact. = COR. 2.-If MV VN, and GN a semi-ordinate, then GM is a tangent, or, if GM be a tangent, GN is a semi-ordinate. PROPOSITION IX. That ordinate of a diameter which passes through the focus, is equal to its parameter. Let GE be any diameter, and RE the semi-ordinate to it, which passes through the focus; then 2RE = 4GD (figure to proposition VII). For RE2 = 4GD · GE (I. 7); but GE = FM = GD (I. 8, Cor. 1). Therefore RE2=4GD2, RE=2GD, and 2RE = 4GD. COR.-If an ordinate to any diameter pass through the focus, the absciss will be equal to the distance of the vertex from the focus. PROPOSITION X. 7 If from any point in the parabola, a parallel to a diameter be drawn to meet an ordinate to the same, the rectangle under the parameter of the diameter and the parallel will be equal to the rectangle under the segments of the ordinate. From any point H, in the curve, let HE be drawn parallel to the diameter VN, to meet its ordinate GQ in E; then For, let the semi-ordinate HK be drawn; then GN2 = P.VN (I. 7), and HK2 = = COR. 1. The parameter of any diameter is to the sum of two semi-ordinates as their difference to the difference of their abscisses. E K N For P HEEG EQ = (GN+HK) (GN-HK). Therefore (Pl. Ge. VI. 16), P:GN+HK=GN-HK: HE, and HE VN-VK. COR. 2.-Straight lines, drawn parallel to a diameter, from points in the curve, to meet any cord, are to one another as the rectangles under the segments of the cord. For let P be the parameter of the diameter; L, L', two of the parallel lines; and R, R', the rectangles under the corresponding segments of the cord, then PL = R, P. L' =R', therefore P·L: P·L'=R: R', or (Pl. Ge. VI. 1) L:LR: R'. COR. 3.-If two parallel cords meet any diameter, the rectangles under their segments will be to each other directly as the parts of the diameter which they intercept from the vertex. For Let S and S' be the segments cut off from the diameter; R and R' the rectangles under the segments of the cords; and P the parameter of the diameter to which the cords are ordinates. Then PSR, and P·S'=R'; therefore R:R'=P·S: P.S'S: S'. PROPOSITION XI. If a diameter be cut by any straight line passing through two points in the parabola, the part intercepted from the vertex will be a mean proportional between the abscisses corresponding to the two ordinates drawn from the same points in the curve. Let G, H, be two points in the curve, and the line GH cut the diameter VN in O; also, let the semi-ordinates GN, HK, be drawn; then VN: VO VO : VK see also figure to proposition X). For VN VK = GN2 : HK2 (I. 7, Cor. 4) = GO2 : HO2 =ON2: OK2; and VN is divided in O and K, so that = V H K N VN, VO, and VK, are in continued proportion. If not, let it be divided in O', so that VN: VO VO': VK ; then by conversion VN: O'N = = VO': O'K; by alternation VN: VO' O'N: O'K, and (Pl. Ge. VI. 22 Cor.) VN2: VO =0′N2: 0′K2. But (Pl. Ge. VI. 20, Cor. 2 and 3) VN: VK = VN2 : VO2, therefore VN : VK = O'N2: O'K2. =ON2: OK2. Hence O'N: O'K = ON: OK; and, by composition, KN : O'N KN: ON; hence O'N ON, and the point O' must coincide with O; therefore VN: VOVO: VK. PROPOSITION XII. = Any two cords which intersect each other in the focus of a parabola, are to one another directly as the rectangles under their segments. Let the cords GH and PQ intersect one another in the focus F, then GH: PQ = GF FH: PF · FQ. For, since GH and PQ are equal to the parameters of the diameters to which they are ordinates (I. 9), GH · VF = GF FH (Ì. 10), and PQ · VF = PF FQ; consequently, GH: PQ=GF FH: PF FQ. P H Q COR.-If two cords intersect in any point, the rectangles under their segments are to one another as the parameters of those diameters to which they are ordinates. For let the rectangles under the segments of the two intersecting cords be denoted by R and R'; and the para meters of the diameters, to which they are ordinates, by P and P'; and L a line drawn from their point of intersection to the curve parallel to the axis; then (I. 10) R: R' =PL:P LP: P' (Pl. Ge. VI. 1). PROPOSITION XIII. If from any point in a parabola, an ordinate be drawn to a diameter, and a tangent to the curve from the same point; and if a circle be described, having the ordinate for a tangent, and the fourth part of the parameter of the diameter through the given point as a cord; a diameter to the parabola passing through the point of intersection of the tangent and the circle, will cut off a fourth part of the ordinate Let EH be a tangent at the point E, and EG an ordinate to the diameter HM of a parabola; EIK a circle, having EK for a cord, and EG for a tangent; and IL a diameter through I; then EG = 4EL. The two triangles EKI, EHM, are similar, because the alternate angles KEI, EHM, are equal, and angles EKI, HEM, are equal (Pl. Ge. III. 32); therefore KE: IE = EH: MH, or KE: IE=EH: 2VH (I. 8). Therefore 2KE · VH=IE EH, but 4KE EN = NV2 (I. 7) = EH2; and A K H B V E M N hence 2IE EH EH2 = EH EH; and therefore 21E EH. Since EH is bisected in I, therefore EL= = LM, or 4ELEG. EXERCISES. 1. Any straight line, drawn from the focus of a parabola to a point in the directrix, is a mean proportional between half the parameters of the diameters which pass through its extremities. 2. If from any point in the parabola, a tangent, semiordinate, and perpendicular, be drawn to meet the same diameter, their squares will be to one another as the parameters of three diameters, that which passes through the point of contact, that which they meet, and the axis. |