CONIC SECTIONS. 



145 



3yper. volve about F, first on one, and then on the other side 

 bola. f t ] ie ax ; s> until in the one case it come to the position 

 "Y"""'' F k, and in the other to FK, it will appear that PN, one 

 side of the constant rectangle FD.PN, will pass through 

 all gradations of magnitude from PL' its greatest value, 

 until it at last vanish, and consequently that PD the 

 other side will increase continually, from P a its least 

 value, until it exceed any assignable quantity. More- 

 over, the hyperbola does not meet either of the lines 

 FK, Yk. 



We may conceive the very same construction to be 

 made at the other focus f, as has been made at F, in 

 both figures; so that, upon the whole, we may draw the 

 following conclusions. 



1. The hyperbola consists of two parts entirely se- 

 parate from each other, and having the directrices be- 

 tween them ; and each is bent at the vertex into two 

 branches, which lie on opposite sides of the transverse 

 axis produced both ways. 



2. If a circle be described on F, either focus, as sta- 

 ted in the proposition, and straight lines KFX, kY x 

 be drawn from its intersections with the directrix 

 through the focus, and produced indefinitely, the part 

 of the hyperbola which is on the same side of the di- 

 rectrix will lie entirely in the space bounded by the 

 directrix and the lines FX, Yx, and it will not meet 

 these lines, and the other part will be entirely contain- 

 ed in the angle y FY, formed by these lines produced 

 the contrary way. 



3. The vertices are the points of the curve nearest 

 to either focus, and of lines drawn from a focus to the 

 curve, those nearer the vertex are less than those more 

 remote. And the curve goes off indefinitely from both 

 foci, and from the prolongation of the axis. 



4. Lines drawn from a focus to the curve, so as to 

 make equal angles with the axes on either side of it, 

 are equal to one another. 



Definitions. 



9. The two parts of an hyperbola, which lie on op- 

 posite sides of the centre, are termed opposite branches 

 qfihe hyperbola. 

 ?• 27. 1 0. If a straight line B b be perpendicular to the trans- 



verse axis at the centre C, and BA, b A, the distance 

 of its extremities from either vertex, be equal to FC 

 the eccentricity ; the line B b is called the Conjugate 

 axis. 



Note. The conjugate axis of the hyperbola is not li- 

 mited in its magnitude by the figure of the curve, as is 

 that of the ellipse ; it is an artificial axis introduced 

 conventionally, in order to preserve the analogy between 

 certain properties of the two curves. 



Con. to Def. 10. The conjugate axis is bisected at 

 the centre of the hyperbola. 



Prop. III. 



The square of the semiconjugate axis is equal to the 

 rectangle contained by the distances of either focus 

 from the vertices of the transverse axis. 



&OT. ForAB'rrAC'+CB* (47.1. E.) and FC 1 ( = AB») 



= AC 1 4- FA. Af (5. 2. E.) therefore BC l = YA.Af. 

 Cor. Let P be the point in which the directrix be- 

 longing to the focus F meets the axis; then, BC 1 = 

 PF.FC. For FC J = PF.FC + PC FC (2. 2. E.) = 

 PF.FC + AC 1 , because AC i = PC.CF (Def. I.); but 

 FC 1 or AB^CB^AC 1 , therefore PF.FC=C£*. 



VOL. VII. PART. I' 



Hyper, 

 bola. 



Definition. 



11. A straight line which meets the hyperbola, and 

 being produced does not cut it, is said to touch the 

 curve, and is called a Tangent. 



Prop. IV. Problem. 



Having given the transverse axis, and a focus, and 

 consequently its directrix, to find the points in which 

 a straight line given by position, perpendicular to the 

 axis, meets the hyperbola. 



Let A« be the transverse axis, F either focus, PQ Fig. 39. 

 the directrix, and L I the line given by position, which 

 meets the axis in G. Draw FM perpendicular to the 

 axis, meeting the curve in M ; draw MPL, meeting 

 the line, given by position in L. On F as a centre, 

 with GL as a radius, describe a circle, meeting LG 

 in D and d ; and these will be two points in the curve. 

 For join FD, Yd, and draw DE, de perpendicular 

 to the directrix ; and because the triangles PGL, PFM 

 are similar, GL : GP : : MF : FP, or, since ED, cf/are 

 each equal to PG, and that FD and Ft? are each equal 

 to LG, FD : DE, also Yd : de : : MF : FP, that is, 

 (Cor. to Def. 1. 3. 4.) in the determining ratio; there- 

 fore D and d are points in the hyperbola, (Def. 1.) 



It appears that the problem can be resolved only 

 when GL=FD is greater than GF. To determine 

 the limits within which this happens, draw A H, a h 

 perpendicular to the axis, meeting PL in H and h ; 

 and join FH, Yh. Then, by similar triangles, PF : 

 FM : : PA : AH : : Pa : a h, but PF : FM : : PA : AF 

 : : P a : aY (Def. I.) therefore AH = AF and ah- a F. 

 Now let GL meet either of the lines FH, Yh in N, 

 then GN=GF ; but when G is any point in the axis 

 A a produced either way, GN or GF is less than GL ; 

 when it is at A or a, then GN or GF=GL, and when 

 it is between A and a, then GN or GF is greater than 

 GL ; therefore the problem will be possible, only when 

 the line L I meets the tranverse axis A a produced. 



Cor. 1. Every straight line perpendicular to the 

 transverse axis, at any point in the axis, produced ei- 

 ther way, meets the hyperbola in two points and no 

 more ; at either extremity of the axis it meets the curve 

 in one point only, and at any point between the ex- 

 tremities of the axis, it falls entirely without both 

 branches of the hyperbole. 



Cor. 2. Every chord perpendicular to the trans- 

 verse axis is bisected by the axis. 



Cor. 3. A perpendicular to the axis at either of 

 its extremities is a tangent. 



Cor. 4. The transverse axis divides the hyperbola 

 into parts exactly alike. 



Prob. V. Problem. 



The transverse axis, and a focus, and consequently 

 its directrix, being given to find the points in which a 

 straight line given by position, parallel to the axis, 

 meets the hyperbola. 



Let EH, the line given by position, meet the direc- p;^ 35, 

 trix in E, and the conjugate axis, or that axis produced 

 in H ; draAV FE from the focus, meeting HC in G. 

 On G as a centre, with a radius, which is a mean pro- 

 portional between EG and GF, describe a circle, meet- 

 ing E H in D and d ; and these will be two points in 

 the hyperbola. 



Join DF, Df, and because EG : GD : : GD : GF and 



