CONIC SECTIONS. 



aa 



Parabola. 5. That Hnc9 drawn from the focus at equal angles 

 ■*— -/-"■"' with the axis on opposite sides of it, and terminating 

 in the curve, are equal to one another. 



Definition. 

 6. A straight line which meets a parabola, and being 

 produced does not cut it, is said to touch the curve, and 

 is called a Tangent. 



Pnop. II. Problem. 

 Having given the focus, and directrix, and conse- 

 quently the axis, to find the points in which a straight 

 line perpendicular to the axis meets the curve. 



Let RS, the line given by position, meet the axis in 



Fig. 63. G, on the focus F as a centre, with a radius equal to 



PG, the distance of the line from the directrix, describe 



a circle to meet the line in D and d, which will be points 



in the parabola. 



For draw DE, tie pei-pendicular to the directrix, 

 then DE=PG=DF, and d e=FG=d.F ; therefore D 

 and d are points in the parabola. (Def. 1 .) 



The construction of the problem requires that FG 

 should not exceed FD, that is PG ; but this condition 

 will manifestly be fulfilled if G be any wherein AX the 

 prolongation of PA, and it would not be fulfilled if G 

 were between P and A : Therefore the line given by 

 position must not be nearer to the directrix than to the 

 focus. 



Con. 1. Every straight line perpendicular to the axis 

 that does not pass through the vertex meets the curve 

 in two points, and no more ; if it pass through the ver- 

 tex, it will meet the curve in one point only, and if it 

 be nearer to the directrix than to the focus, it will not 

 meet the curve at all. 



Cor. 2. Every chord perpendicular to the axis is bi- 

 sected by the axis. 



Cor. 8. A perpendicular to the axis at the vertex is 

 a tangent. 



Prop. III. Problem. 



Having the focus and directrix, and consequently the 

 axis, to find the point in which a line parallel to the 

 axis meets the parabola. 



Let ER, the line given by position, meet the direc- 

 trix in E. Draw FE to the focus, and bisect EF at 

 O, by a perpendicular OT; anci because PER is aright 

 angle, the angle OER is not a right angle, and there- 

 fore the lines ER, OT will meet at a point D, which 

 will be a point in the parabola. For the triangles FOD, 

 EOD are manifestly in all respects equal (4. 1. E.), 

 therefore DF=DE ; and hence D is a point in the pa- 

 rabola (Def. 1.) 



Cor. Every straight line parallel to the axis meets 

 the parabola in one point and no more. For besides D, 

 no other point can be found in ER that is equally dis- 

 tant from E and F. 



Prop. IV. 

 If from any point Q in the directrix a straight lin e 

 Fig. 05. be drawn to meet the parabola in D, and LF I a paral- 

 lel to the directrix through the focus in H ; and if ano- 

 ther straight line be drawn to D from the focus, a per- 

 pendicular QK, drawn to this last line from Q, shall 

 be equal to FH, the segment cut off by the former 

 from the line FL. 



Draw DE perpendicular to the directrix, and pro- 

 duce DF to meet it in I. The triangles DFH, DIQ 

 are manifestly equiangular (29. 1. E.) as also the tri- 

 2 



*Ig. 6' 



angles IED, IKQ (32. 1. E.) therefore, DF:FH ParahdU. 



(: : DI : IQ) : : DE : QK; but DF=DE, (Def. 1.) there- — *\ 



fore FH=QK. 



Con. 1. If the lines FD, QH have such a position 

 that FH is equal to QK the perpendicular drawn from 

 Q to FD ; then if FD and QH are not parallel, they 

 will meet at D, a point in the parabola. 



Cor. 2. If QF be drawn to the focus, and in LF /, 

 which is parallel to the directrix, there be taken FL, 

 and F/ each equal to FQ, then, any straight line drawn 

 from Q to meet the parabola wiU either pass through 

 the points L, /, or between these points. For QK can- 

 not exceed QF, therefore FH cannot exceed FL. 



Cor. 3. Any straight line drawn from Q, to pass be- 

 yond the limits L i will be entirely without the para- 

 bola. 



Prop. V. Problem. 



The focus and directrix being given, to find the 

 points in which a straight line not perpendicular to the 



axis meets the curve. 



Case 1. Let the line given by position be XY, which 

 passes through F the focus. Draw FQ perpendicular 

 to XY, meeting the directrix in Q, and in a line pas- 

 sing through F, parallel to the directrix, take FL and 

 F I each equal to FQ ; and join QL, Q/, then as the 

 angles FQL, FQZ cannot be right angles (5. and 17. 

 1. E.) the lines QL, Q/ must meet XY in two points 

 D, d, which will be in the parabola. (Cor. 1.) 



Case 2. Next, let the line given by position meet 

 the directrix in Q (Fig. 67.) and the parallel LF/ in 

 H, so that FL and F I being taken each equal to FQ, 

 the point H may fall between L and /. On Q as a 

 centre, with a radius equal to FH, (which is less than 

 FL or FQ,) describe a circle ; draw FK, F k tangents 

 to that circle at K and lc ; and if neither of these is 

 parallel to QH, they will meet it in D, d, which will 

 be two points in the parabola, as is evident from Cor. 

 1. Prop. 4. by joining QK, Qk. 



If one of the lines, as F k, were parallel to QD, so 

 that there was only one intersection, then as FH = Q k, 

 the figure It QHF would, in that case, be a rectangle, 

 and so QH would be perpendicular to FL, or parallel 

 to the axis, a conclusion which agrees with what has 

 been already shewn in Prop. 3. 



Case 3; Lastly, let the line Q I, given by position, 

 pass through /, either of the two points L, I, determined 

 as above. Draw FD' perpendicular to FQ, and be- 

 cause FQ/ is not a right angle (7. 6. and 17- 1. E.) the 

 lines Q /, FD' must meet, and D', their intersection, is 

 a point in the parabola ( i . Cor. 4.) 



Cor. 1. The points L, / being supposed determined 

 as in the proposition, every straight line drawn from 

 Q to pass between L and /, either meets the curve in 

 one point only, or in two points and no more ; for the 

 number of intersections cannot exceed the number of 

 tangents to the circle, as has been explained in Sect. 

 II. Prop. 7. 



Cor. 2. A straight line drawn from Q, through L, 

 or /, meets the curve in one point only, and is a tangent. 

 The reason is the same as has been given in the like 

 case of Prop. 7- Sect. II. 



Cor. 3. Every straight line which meets a parabola 

 cuts it in one point only, or in two jjoints, and no more, 

 or it touches it, and these are all the varieties that can 

 happen. 



Cor. 4. If two lines FD, FQ, which contain a right 

 angle at the focus, meet the curve in D, and the di- 



es. 



Fig. C7. 



