only (as in the parabola), the portion of it within the curve is called the principal diameter or axis of the parabola. 6. The middle of the transverse diameter of the ellipse or hyperbola is called the centre of the curve, and a line drawn through the centre at right angles to that diameter is the conjugate diameter. 7. Any line drawn through the centre of a conic section, and limited by the curve, is called a diameter of the conic section. In the case of the parabola, the centre is infinitely distant, and therefore all its diameters are parallel to the principal diameter. 8. The points of section of the diameters with the curve are called the vertices of those diameters. 9. A tangent to a conic section is a straight line which meets the curve, but being produced, does not cut it. 10. If a tangent and a diameter of a conic section pass through the same point, they are said to be conjugate to one another; and any lines drawn parallel to these are said to be conjugate to one another. Of two lines drawn in this way, if both pass through the centre of a conic section, they are said to be conjugate diameters; and if one be a diameter, and the other a chord of a conic section, the chord is called an ordinate of that diameter. Also the part of the diameter between the vertex and ordinate is called the abscissa; and the abscissa and ordinate, when spoken of together, are called coordinates of the point or points. 11. The subtangent is that portion of the principal diameter of a conic section intercepted between the tangent and the ordinate drawn from the same point in the curve. 12. The normal is that portion of the perpendicular drawn to the tangent from the point of contact, which is intercepted between that point and the transverse axis; and the subnormal is the portion of the transverse axis intercepted between the normal and the ordinate at the same point. 13. The parameter, or (as it is sometimes called) the latus rectum of any diameter, is the third proportional to that diameter and its conjugate in the ellipse and hyperbola, and to any abscissa and its ordinate in the parabola. [This is not a mere arbitrary definition, as might be supposed, but a term suggested by a property of the curve. Its demonstration is given in a subsequent part. By the term parameter in general is meant, some constant quantity of the curve. In the case of the circle, for instance, the radius may be considered as the parameter]. 14. The focus of a conic section is that point in the transverse diameter at which the double ordinate is equal to the parameter of the transverse diameter; and the distance between the centre and focus is called the eccentricity. 15. If a third proportional (estimated from the centre in the ellipse and hyperbola upon the transverse axis) be taken to the eccentricity, and the semi-transverse diameter, then the perpendicular to the transverse axis drawn through the extremity of the third proportional is called the directrix. In the parabola, the directrix is at the same distance from the vertex that the focus is, [It will be seen, when we come to the investigation of properties, that the directrix has a remarkable relation to each curve. For instance, the distance of any point in the curve from this line has a constant ratio to the distance of the same point from the focus. This property of the conic sections is made the foundation of the coordinate method given in the first volume. The constant ratio has been named the determining ratio.] THE PARABOLA. PROPOSITION I. In the parabola, the abscissas are to one another as the squares of their ordinates. Let MVN be the transverse plane, iAI the parabolic sectional plane, AH the transverse axis, KGL, MIN two sections of the cone perpendicular to its axis. Also let KL, MN be the intersections of KGL, MIN with the transverse plane, and GFg, IHi those with the sectional plane. Then because KGL, MIN are sections perpendicular to the axis of the right cone, they are circles (Pls. 1. 24); and since the transverse plane passes through the axis of the cone, it passes through the centres of these circles, and therefore the lines KL, MN are diameters of the circles KGL, MIN. M K H Again, because the plane MVN passes through the axis of the cone, it is perpendicular to the base, and each of its parallel sections KGL, MIN (Pls. II. 16). It is also perpendicular to the sectional plane IAi (Def. 4); hence the two planes KGL, IAi being each perpendicular to the plane MVN, their common section Gg is perpendicular to the same plane (Pls. 11. 18), and consequently to every line in it, as AH, KL. In a similar way it is shown that Ii is perpendicular to AH and MN. Also, since KL is a diameter of the circle KGL, and Gg is perpendicular to this diameter, Gg is bisected in F. Similarly Ii is bisected in H. Now by the similar triangles FAL, HAN, and the equal lines KF, MH (AH being parallel to VM by Def. 3), we have AF: AH:: FL: HN:: KF.FL: HM. HN. But by a property of the circle (Euc. 111. 35), And as AF, AH are obviously the abscissas, and FG, IH the ordinates of the points G and I in the curve, the property is consequently established. COR. 1. Let B and P be any two points in the parabola PAP', and BF, PM their semi-ordinates; then by the proposition, AF:AM:: BF2 : PM2, or AF: BF2 :: AM: PM*. Hence, if we denote the third proportional to AF and BF, or to AM and PM, by p, it will be obvious that p will be constant for every point in the curve; and therefore we have at any point Q, NQ2=p. AN; that is, if we call Р the parameter of the principal diameter, the rectangle under the abscissa of any point in the parabola and the parameter of the axis is equal to the square of the semiordinate of that point. This is one of the properties to which reference is made in Def. 13. = A COR. 2. Let AF be of such magnitude that BF = p; then by Cor. 1, AF tp. Whence (Def. 14), the abscissa of the ordinate which passes through the focus of a parabola is equal to one-fourth of the parameter of the axis. AE COR. 3. Let EX be the directrix of the parabola PAP'; then (Def. 15), FA (Cor. 2) p. Wherefore, the distance between the focus and directrix of a parabola is equal to one-half the parameter of the axis. COR. 4. Let QQ' be any other ordinate to the axis of the parabola, and PC a line parallel to the axis, meeting the curve in P and the ordinate QQ'in C. Then by the proposition, p. AN -p. AM = NQ MP2 = (NQ+ MP) (NQ – MP) = Q'C. QC; or p. PC = Q'C. QC. Ilence, if a double ordinate of the parabola be divided into two segments by a line parallel to the axis, the rectangle under the two segments is equal to the rectangle under the dividing line and the parameter of the axis, the dividing line being limited by the ordinate and curve. PROPOSITION II. In the parabola, the line drawn from the focus to any point in the curve, is equal to the perpendicular drawn from the same point to the directrix. (See last Figure.) Let P be any point in the curve, PX a perpendicular on the directrix, and F the focus, of a parabola; then PF = PX, For PM being a perpendicular on the axis AM, we have (Euc. 11. 4), PX2 = EM2 = EF2 + FM2 + 2EF.FM But by Def. 15, and Prop. 1., Cor. 3, EF (Prop. 1.), p AM PAM+FM2 = = = = = 2 AF, and also MP. Whence PX = p(AF + FM) + FM2 PM2 +FM2 or PX = PF. = PF2; COR. A line drawn from the vertex A perpendicular to the axis, is a tangent at the vertex. For any point in this perpendicular is evidently at a greater distance from the focus F than A is, and therefore it is at a greater distance from the focus than from the directrix. Consequently by the proposition, this perpendicular does not meet the curve again; and hence (Def. 9) it is a tangent to it. SCHOLIUM 1. The property established in this proposition suggests the following simple method of constructing a parabola whose principal parameter is given : = = Make AF AE one-fourth of the parameter (Prop. 1., Cor. 2), and in EAF produced, take any point M; with F as centre and radius EM, describe a circle, intersecting a perpendicular PMP' to EAF produced, in P and P'; then will P and P' be two points in the curve. Other points are found in a similar way. SCHOLIUM 2. The property itself is the determining or constant ratio, for the parabola, to which reference is made in Def. 15. PROPOSITION III. If a line be drawn to bisect the angle made by two lines, one of which is drawn from a point in the parabola to the focus, and the other from the same point perpendicular to the directrix, that line will be a tangent to the parabola at that point. Let P be a point in the parabola, F the focus, EX the directrix, perpendicular to which PX is drawn; then the line PT drawn to bisect the angle FPX, will be a tangent at P. For since P is a point in the curve, TP meets the curve; and if it be not a tangent, it will meet the curve again in some other point K. Then, also, K is a point in the parabola. Join KF, and draw KL perpendicular to the directrix, and join XK. Then because K is a point in the curve, and KL is perpendicular to the directrix, KF = = L E K F M N KL (Prop. 11.); and since by the same Prop., XP = PF, and by hypothesis the angle XPK = FPK, and KP common to the triangles XPK, FPK, the base FK, therefore, is equal to KX. But it has been shown that FK KL, and therefore KL = KX, the less to the greater, which is impossible. Whence the point K is not in the parabola. And in the same way it may be shown that no other point of PT is in the parabola except P. Hence (Def. 9) PT is a tangent to the parabola. = = COR. 1. Since the line XP is parallel to FT, the angle XPT PTF; but by hypothesis, XPT FPT. Hence FTP = FPT, and FT = FP; that is, the point of contact of a tangent to a parabola and the point of intersection of the tangent and axis produced, are equally distant from the focus. COR. 2. Draw the ordinate PM; then AF + AT = PX = FT = FP AM; AF AM. Hence AT or that is, the subtangent of a parabola is bisected in the vertex (Def. 11). COR. 3. Draw the normal PN. Then (Def. 12), TPN is a right angle, and therefore (Euc. VI. 8), TM: MP :: MP: MN, TM.MN == MP2. But (Prop. 1., Cor. 1), p. AM P.AM And since by last Cor., TM VOL. II. = TM. MN. = MP; hence = 2 AM, conse Q quently MN p.; that is, in the parabola, the subnormal is equal to half the parameter of the axis. COR. 4. From the focus F draw FH perpendicular to the tangent PT, and join A and H. Then because FT = FP (Cor. 1), and MA =AT (Cor. 2), the triangles PTM and HAT are similar, since the perpendicular FH evidently bisects PT. Wherefore AH is the tangent at the vertex A. Hence, a perpendicular from the focus of a parabola on any tangent intersects that tangent in the tangent from the vertex. COR 5. Also by Euc. vi. 8, FT: FH:: FH: FA; that is, if a straight line be drawn from the focus of a parabola perpendicular to the tangent at any point, it will be a mean proportional between the distances of the focus from the vertex and the intersection of the tangent and axis. SCHOLIUM 1. From the property of tangents to the parabola established in this proposition, the term focus is given to the point F. For by a property of optics, rays of light which proceed parallel to the axis of a parabola and fall on a polished surface whose figure is that produced by the revolution of the parabola about its axis, are reflected to this point. SCHOLIUM 2. It would be easy to extend this series of properties of the parabola if the limits of this course admitted of such extension. All those properties relative to the principal axis and its coordinates hold for any diameter and its coordinates. The following is one of the most important of these. PROPOSITION IV. The abscissas of any diameter of a parabola are to one another as the squares of their ordinates. Let EH be any ordinate to the diameter CV, so that it is parallel to the tangent CT at the point C; CM the corresponding abscissa; AI the axis; and F the focus of a parabola. From E and C draw EG, CD perpendicular to the axis, and from the focus F draw FY perpendicular to the tangent CT; also join FC, and produce EG to meet CV in S. H M Then it will be obvious by Prop. EM: ES2:: FC2: FY2. But (Prop. III., Cor. 5), since FT = FC, FC: FY:: FY: FA, or FC: FA:: FC: FY. Wherefore, EM2: ES :: FC: FA. . . Again, by the similar triangles CTD, ESM, ES: SM:: CD: DT:: CD2: CD. DT. And since (Prop. 1., Cors. 1, 2), CD 4 AF. AD, and (Prop. III., Cor. 2), DT=2 AD; we have = ES: SM:: 4 AF. AD: 2 CD. AD:: 4 AF: 2 CD, or, 4 AF. SM = 2 CD. ES |
