Hence, by division and similar triangles, remembering that QP = RC, we have But by Prop. IX., Cor. 2, and duplicate ratio, = Now because C is the middle of ED, CD2 – CQ2 EQ . QD ; consequently EQ . QD : QP2 : : CD2 : CH2 :: ED2: HG; that is, the square of the diameter CD is to the square of its conjugate HG as the rectangle under the abscissas EQ, QD, is to the square of the ordinate PQ. This is the general case of the property established in Prop. II. COR. 1. The rectangles of the abscissas of any diameter are as the squares of the corresponding ordinates. THE HYPERBOLA. PROPOSITION I. The squares of the ordinates of the transverse diameter are to one another as the rectangles of their abscissas. As in the parabola and ellipse, let AVB be the transverse plane; AGIB the hyperbolic sectional plane which meets both sheets of the cone (Def. 3); AB the transverse axis which meets the two sheets of the cone in A and B; KGL, MIN, any 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 it may be shown, as in the parabola (Prop. 1.), that KL, MN are diameters of the circles KGL, MIN, and that the transverse diameter AB bisects Gg, Ii, perpendicularly in F and H. Hence by the similar triangles AFL, AHN, and KFB, MHB, K WA T AF: AH:: FL: HN, and FB: HB:: KF : MH. Whence AF. FB: AH. HB:: FL. KF: HN. HM. But by Euc. III. 35, FG; and HN. HM = HI'. Wherefore, AF. FB: AH. HB:: FG: HI; that is, the squares of the ordinates FG, III, are to one another as the rectangles of their corresponding abscissas. The complete hyperbola, therefore, is situated on both sheets of the cone, and is composed of two separate and infinitely extended branches (Def. 1). Those branches, moreover, are divided by the transverse diameter into two parts, which are symmetrical with respect to that diameter. PROPOSITION II. The rectangle under the abscissas of any point in the hyperbola is to the square of the corresponding ordinate as the square of the transverse diameter is to the square of its conjugate. (See the Fig. of last Prop.) Let a circular section of the cone meet the middle of the diameter AB in C; and let Ca be a tangent from C to this circle. Then Ca is the conjugate axis to AB. Let WT be the diameter of this circular section; then by the similar triangles AFL, ATC, and FBK, BWC, we have AF: AC:: FL: TC, and FB: BC :: FK: WC. Hence, AF. FB: AC. CB:: FL. FK: TC . WC. But since AC CB, AC. CB AC, also (Euc. 111. 35, 36) FL. FK FG2 and TC. WC 1=3 Ca. Wherefore = = = AC: Ca2 AF. FB: FG2, or AB2: 4Ca2 :: AF. FB: FG2; that is, the square of the transverse diameter is to the square of its conjugate as the rectangle under the abscissas of the point G is to the square of the ordinate of the same point. COR. Take p, a third proportional to AB and 2Ca; then by the property of duplicate ratio, AB: p:: AB: 4Ca2. But by the proposition AB2: 4Ca2:: AF. FB: FG2. Wherefore, AB :p :: AF. FB: FG'. The quantity p which remains constant for different points in the curve is the parameter of the transverse diameter AB (Def. 13). Wherefore the transverse diameter of the hyperbola is to its parameter as the rectangle under the abscissas of any point is to the square of the corresponding ordinate. PROPOSITION III. In the hyperbola, the square of the distance of either focus from the centre is equal to the sum of the squares of the semi-diameters. Let AB be the transverse diameter of the hyperbola, ab its conjugate, F and ƒ the foci, C the centre, and GF an ordinate at the focus F. Then by Prop. 2 and Euc. 11. 6, CA: Ca2:: AF. FB: GF:: CF2 CA: GFs. But F being the focus, GF=p (Def. 14), and therefore by Def. 13 AC2: Ca2:: Ca2 : ¦ p2 or GF2. Hence CF CA 1=3 Ca2, or CF2 1= CA' + Ca'. D COR. Join Aa; then, by Euc. 1. 47, and the Prop., Aa2 = = AC+ Ca2 CF; hence the distance of either vertex of the conjugate diameter from either vertex of the transverse is equal to the focal distance of the centre. PROPOSITION IV. If in the hyperbola a fourth proportional be taken to the semi-transverse diameter, the eccentricity, and the distance of an ordinate to any point in the curve (all estimated from the centre along the transverse axis), then the parts of the transverse axis intercepted between the extremity of the fourth proportional and the extremities of the transverse are respectively equal to the focal distances of that point. (See the Fig. of last Prop.) Let E be any point in the hyperbola, F and ƒ the foci, and CP a fourth proportional to CB, Cf, and CD; then the segments AP, BP, of the transverse axis, intercepted between P and the extremities A and B of the transverse diameter, will be respectively equal to EF, Ef, the focal distances of the point E. For by construction, CB2: Cf2:: CD: CP2; hence since (Prop. 111.) Cƒa — CB2 CB2: Ca2:: CD2 : CP2 CD2. But (Prop. II.), Proceeding with this proportion as in the analogous case of the ellipse (Prop. IV.) we get EF AP, and Ef = BP. COR. The difference of the lines drawn from the foci to any point in the hyperbola is equal to the transverse diameter. PROPOSITION V. The distance of the directrix from any point in the hyperbola has to the distance of the focus from the same point a constant ratio. Let NL be the directrix with respect to the focus F, and N'L' that with respect to the other focus f; then E being any point in the curve, the ratio of FE to EL, or of ƒE to EL' is equal to the constant ratio of CF to CA (C being the centre). For take in the transverse axis AB, CF : CA :: CA: CN, and draw NL perpendicular to the axis; then NL is the directrix with respect to the focus F (Def. 15). Also, as in the last Prop., take CP a fourth proportional to CA, CF, and CD. Then we have Hence CF: CA:: CA: CN, and CF : CA :: CP: CD. CA: CN :: CP : CD, or CA: CP :: CN: CD. AP CA: DN: CN; or since AP = EF (Prop. IV.), And if N'L' be the directrix with respect to the other focus f, it may be shown in a similar way, that Ef: EL':: CF : CA. PROPOSITION VI. The straight line which bisects the angle adjacent to that which is contained by two straight lines drawn from any point in the hyperbola to the foci, is a tangent to the curve at that point. The demonstration of this, which is exactly similar to the analogous case for the ellipse (Prop. vI.), is left for the student's exercise. THE ASYMPTOTES. a * DEF. 16. If through A, one of the vertices of the transverse diameter of the hyperbola, a straight line KAN be drawn, equal and parallel to a b the conjugate diameter, and bisected in A by the transverse diameter; the straight lines CK, CN drawn through the centre, and the extremities of that parallel, are called asymptotes to the hyperbola. The asymptotes also of two opposite hyperbolas are common to both. 8 There are many remarkable properties connected with the asymptotes to the hyperbola; but the limits of this course do not admit of our entering upon them. It may be stated, however, that the hyperbola and its asymptotes, when produced, continually approach to each other, but meet only at an infinite distance. (See Vol. I., p. 306). CONJUGATE HYPERBOLAS. DEF. 17. If ab be taken for the transverse diameter, and AB for its conjugate, of other two hyperbolas in the spaces KCL, MCN (a and b being the vertices of those diameters); then those two other hyperbolas are said to be conjugate to the former. When all four are mentioned, they are called conjugate hyperbolas. Additional Problems and Theorems for Exercises. 1. Prove that if any two diameters of a parabola be produced to meet a tangent to the curve; the segments of the diameters between their vertices and the tangent are to one another as the squares of the segments of the tangent intercepted between each diameter and the point of contact. 2. A parabola is given in position, to find its directrix and focus. 3. If any chord be drawn parallel to the tangent of a parabola, to meet the curve in two points; and if from its extremities and from the point of contact ordinates be drawn to the axis, then double the ordinate of the point of contact will be equal to the sum or difference of the other two, according as they are situated on the same or different sides of the axis. 4. If from the point of contact C of a tangent to a parabola, any chord CL be drawn, and another line parallel to the axis to meet the chord, curve, and tangent, in K, E, I; then IE: EK :: CK: KL. 5. If from the point of contact of a tangent to a parabola, two lines be drawn to the vertices of any two diameters, each to intersect the other diameter; then the line joining the two points of intersection will be parallel to the tangent. 6. If the chord of contact of two tangents to a parabola pass through the focus, the tangents will intersect at right angles to one another in the directrix. 7. Let TP, tp, be tangents to a parabola at the points P and p, and PM, pm, ordinates on the axis from the same points; then (1) tan TPM: tan tpm :: PM: pm; - PM. 8. Show how to draw a tangent and normal to a given parabola from a given point without the curve. 9. Prove that any chord of an ellipse drawn through the focus is so divided in that point that four times the rectangle contained by its segments is equal to the rectangle contained by the chord and the parameter of the transverse diameter. 10. Let a tangent to an ellipse intercept the four perpendiculars on the axis, from the centre, the two extremities of the axis, and the point of contact; then will those intercepted perpendiculars be proportionals. |