CONIC SECTIONS. 



For the right line which bisects all 

 the parallel passes through the centre ; 

 and therefore it must coincide with the 

 line that bisects one of the parallels, 

 and is drawn through the centre. ' 



PROP. xv. 



Fig. 27. All the diameters of a parabo- 

 la are parallel to one another. 



Let B C be a diameter of a parabola bi- 

 secting the lines D E and F G : take any 

 point L within the parabola, and draw M 

 N through it parallel to D E or F G, and 

 terminated by the curve : then B C will 

 bisect M N ; and as this is true, however 

 remote from the lines D E and F G the 

 line M N is drawn, it follows that the di- 

 ameter B C cannot meet the curve in 

 more than one point : and the same thing 

 may be shewn of every other diameter as 

 P Q. But all those right lines are paral- 

 lel to one another which cut a parabola in 

 one point only. (Cor. 2. Def. 6.) 



Cor. A right line parallel to a diame- 

 ter of a parabola, which bisects one right 

 line, terminated by the parabola, will bi- 

 sect all other right lines parallel to the 

 former and terminated by the parabola. 



Def. 11. A diameter of two opposite 

 hyperbolas, which is terminated by the 

 two curves, is called a tranverse diame- 

 ter : and a diameter which meets nei- 

 ther of the curves is called a second 

 diameter. 



Def. 12. A vertex of a diameter is a 

 point where the diameter meets the 

 conic section. 



The magnitude of a diameter, that 

 meets a conic section or opposite sec- 

 tions in two points, is the line between 

 the two vertices. 



Def. 13. A right line, not passing 

 through the centre, terminated by a conic 

 section, or opposite sections, and bisected 

 by a diameter, is said to be ordinately 

 applied to that diameter : or it is called 

 a double ordinate, and the half of it an 

 ordinate. 



PROP. XVI. 



Fig. 28. A right line drawn from a 

 vertex of a diameter of an ellipse, or a 

 parabola, or from the vertex of a trans- 

 verse diameter of a hyperbola, so as to 

 be parallel to a line ordinately applied to 

 that diameter, is a tangent of the curve. 



Fig. 28. Let F H be a diameter of an 

 ellipse or a parabola, or a transverse di- 

 ameter of a hyperbola, and R S T, a line 

 ordinately applied to that diameter ; then 



F M, drawn from a vertex of the diame- 

 ter, so as to be parallel to R T, is a tan- 

 gent of the curve. For, if F M be not a 

 tangent, it will cut the section again in 

 another point (Cor. 2 Def. 8,) let it cut 

 the section again in K, and bisect F Kin 

 I. Then, if a diameter of the section be 

 drawn through I, that diameter would 

 bisect RT parallel to F K,Pr. 15. There- 

 fore R T would be bisected by two differ- 

 ent diameters ; viz. by the diameter F H, 

 and by that drawn through I. But, in 

 the ellipse and hyperbola, all the dia- 

 meters pass through the centre ; and in 

 the parabola, they are all parallel to one 

 another ; therefore two diameters of a 

 conic section will cut every straight line 

 (which does not pass throng* the cen- 

 tre of the ellipse and hyperbola) in two 

 different points. Therefore R T cannot 

 be bisected by two different diameters. 

 Therefore F M, parallel to R T, does not 

 cut the curve again ; that is, F M is a tan- 

 gent of the conic section. 



Cor. 1. If R T be ordinately applied 

 to the diameter F H, it is parallel to a 

 tangent, F M, at a vertex of that diameter. 



For there cannot be two tangents of a 

 conic section at the same point of the 

 curve. 



Cor. 2. All right lines ordinately ap- 

 plied to the same diameter of a conic sec- 

 tion are parallel to one another. 



For they are all parallel to a tangent at 

 a vertex of that diameter. 



PROP. XVII. 



Fig. 29. A right line D E terminated 

 both ways by the curve of a conic section, 

 and parallel to a tangent F H, is ordi- 

 nately applied to the diameter B C drawn 

 through the point of contact B. 



Take B F and B H, in the tangent on 

 opposite sides of the point of contact, 

 equal to one another, and of such a magni- 

 tude that lines drawn through F and H 

 parallel to the diameter B C may cut the 

 curve in K and L : join K L. It is plain 

 that K L is bisected by B C : therefore K 

 L is parallel to the tangent F H (Cof. 1, 

 16.); and consequently it is also parallel 

 to D E (30. 1. E.) ; therefore D E is bi- 

 sected by the same diameter which bisects 

 K L (Co'r. 14,) 



Def 14. Two diameters of an ellipse, 

 or of opposite hyperbolas, that are mutu- 

 ally parallel to one another's ordinate, are 

 called conjugate diameters. 



Cor. It is plain that two conjugate 

 diameters of opposite hyperbolas can- 



