CONIC SECTIONS. 



139 



Wlipse. tance of KF from the point Q), the determining ra- 

 y -—-V"-"' tio, the lines QH, KFD will meet at a point D in the 

 ellipse : For then, FD : DE : : FH : QK, that is in the 

 determining ratio. 



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

 LF/ parallel to the directrix, there be taken FL, F / in 

 contrary directions, and each to QF in the determining 

 ratio, then every straight line drawn from Q to meet 

 the ellipse, will either pass between the points L, /, 

 or through these points. For since by hypothesis 

 FQ : FL : : KQ : FH ; and since KQ cannot exceed 

 FQ, therefore FH cannot exceed FL. 



Con. 3. Any straight line drawn from Q to cut the 

 line hi beyond the limits L, / falls without the ellipse. 



Prop. VII. Problem. 



Having given a focus, its directrix, and the deter- 

 mining ratio, to find the points in which a straight line 

 given by position, and not parallel to the directrix, 

 meets the ellipse. 



•ig. 8. Case 1. Let XY, the line given by position, pass 



through F the focus ; draw FQ perpendicular to XY, 

 meeting the directrix in Q ; and in LF/, a line passing 

 through the focus parallel to the directrix, take FL 

 and FY, each to KQ in the determining ratio; and 

 because P'L anclF/ are each less than FQ, the angles 

 FQL, FQ / are not right angles (Geometry), therefore 

 the lines QL, Ql will always meet the line XY in two 

 points D, d, which will be points in the ellipse, as is 

 evident from Cor. 1 . Prop. 6. 



'ig. 9. Case 2. Next, let the line given by position meet 



tfie directrix in Q, and I, / the line passing through the 

 focus parallel to the directrix in H; so that QF being 

 drawn to the focus, and FL and F / taken each to FQ 

 in the determining ratio, the point H fails between the 

 focus, and either of the points L, I. In QF take QN, 

 so that FH may have to QN the determining ratio, 

 then FH : QN : : FL : QF ; but FH is less than FL, 

 therefore QN is less than QF : On Q as a centre, with 

 QN as a radius, describe a circle, and draw FK, F k 

 tangents to it, also draw FO perpendicular to QH ; 

 then, as FH is less than KQ, FO will be much less 

 than KQ, or k Q ; therefore the lines KF, £F cannot 

 be parallel to QO, and they must, if produced, meet it 

 in two points Dfd, which will be points in the ellipse, 

 as is evident from Cor. 1. Prop. b". 



S 9. Case 3. Lastly, let the line given by position meet 



the directrix in Q (Fig. 9.), and be so situated, that 

 QF being drawn to the focus, and FL and F / each 

 taken to QF in the determining ratio, the line passes 

 through L, either of the extremities of the line hi; 

 then it will appear, as in the first case, that FQL can- 

 not be a right angle; therefore, FD' being drawn per- 

 pendicular to FQ, will meet QL in D', a point in the 

 ellipse, Cor. 1. Prop. 6. 



Cor. 1. The points L, / being determined, as in the 

 second and third cases, any straight line drawn from 

 Q, to pass between them, will meet the ellipse in two 

 points, and no more : For it has been shewn, that the 

 tangents FK, F/ t - must always meet the line given by 

 position in two points D, d ; and as these are evidently 

 the only lines which can be drawn through F, so that 

 FH may have to QK, Q/:, (their distances from Q) 

 the determining ratio, there can be no more than two 

 intersections of the line QH and the ellipse. 



Cor. 2. If from D, r/, the intersections of a straight 

 line and an ellipse, straight lines DF, </F be drawn to 

 a focus, these make equal angles with QFR, a line- 



drawn through the focus from Q the point in which El'ip 

 the intersecting line meets the directrix. For the angles "" 

 dFQ is equal to KFQ or DFR. 



Cor. 3. The points L, / being determined as in the 

 proposition, a straight line drawn from Q through ei- 

 ther of these points is a tangent to the ellipse. For in 

 this case only one line FD' can be drawn through F, 

 so that FL has to FQ its distance from Q, the deter- 

 mining ratio, and it appears that the ellipse lies wholly 

 on one side of FL, (Cor. 3. 6.) therefore FL is a tan- 

 gent. (Def. 10 ) 



Cor. 4. If a straight line meet an ellipse, it will 

 either touch it, or cut it in two points, and in no more. 

 This follows from the Prop, and Prop 4. and 5. 



Cor. 5. If two straight lines FD, FQ (Fig. 10.) Fijf.'lft 

 which contain a right angle at the focus, meet the 

 curve and the directrix in D and Q ; the straight line 

 DQ, joining these points, is a tangent to the ellipse at 

 D ; and only one tangent can be drawn to the curve 

 at that point. It is evident from the construction of 

 Case 3. and Cor. 3. that the line DQ is a tangent. 

 To prove that there is no other tangent at D, draw any 

 other line DQ' through D ; if this line be parallel to 

 the directrix, it will cut the curve (Prop. 4.) but if it 

 be not parallel, let it meet the directrix in Q', and FH 

 a parallel to the directrix passing through the focus in 

 H. Draw Q'K perpendicular to FD ; join Q'F ; and 

 take FL to Q'F in the determining ratio ; and because 

 D is in the ellipse, FH has to Q'K the determining ra- 

 tio (6.) therefore FQ' : FL : : Q'K : FH, but FQ' is 

 greater than Q'K, therefore FL is greater than FH ; 

 hence the line Q'H must meet the ellipse in two points 

 by Cor. 1 . of this proposition. 



Cor 6. Tangents DQ, d Q (fig. 8.) at the extremi- 

 ties of any focal chord D d, and a perpendicular FQ V ''S- 11 * 

 to that chord at the focus, meet at a point Q in the di- 

 rectrix. 



Prop. VIII. 



A tangent to the ellipse makes equal angles with 

 straight lines drawn from the point of contact to the foci. 



If the tangent be at either extremity of the tran- 

 verse or conjugate axis, the truth of the proposition is 

 manifest ; in any other case let it touch the curve at 

 D, and meet the directrices in Q and q; draw DF, D/* 

 to the foci, and ED e perpendicular to the directrices. 

 The triangles DEQ, D c q. axe manifestly equiangular^ 

 therefore DQ : DE : : D q : D e (l. 6. E :) but DE : 

 DF : : D e : D/ (Def. 1 .) therefore ex. aeq. DQ : DF : : 

 D§ : Dy (22. 5. E.) hence it appears that the triangles 

 DFQ, Djq, which have the angles at F imdj right 

 angles, (Cor. 5. 7.) have the sides about one of the re- 

 maining angles in each proportionals, therefore they are 

 equiangular, (7. 6. E) and have the angles QDF, q Df 

 equal. 



Definition 1 1, 

 A straight line passing through the centre, and ter- 

 minating both ways in the ellipse, is_c;iiied a digme-^ 

 let, and the points in which it meets the ellipse are 

 called its veitccif. 



Prop. IX. 



Every diameter is bisected at the centre. 



Fig. 13; 



h, and join H h, D C, d C : Then DH = H d ', (2. Cor. 

 5.) = C h (34. 1. E.) and HC zzd'hzzhd (2, Cor. 4.) 



