File:A geometrical treatise on conic sections, with numerous examples. For the use of schools and students in the universities. With an appendix on harmonic ratio, poles and polars, and reciprocation (14754399496).jpg

Identifier: geometricaltreat00drewuoft (find matches)
Title: A geometrical treatise on conic sections, with numerous examples. For the use of schools and students in the universities. With an appendix on harmonic ratio, poles and polars, and reciprocation
Year: 1887 (1880s)
Authors: Drew, William Henry
Subjects: Conic sections
Publisher: London Macmillan
Contributing Library: Gerstein - University of Toronto
Digitizing Sponsor: MSN

View Book Page: Book Viewer
About This Book: Catalog Entry
View All Images: All Images From Book
Click here to view book online to see this illustration in context in a browseable online version of this book.

Text Appearing Before Image:
e in thecircle E Qe and the cutting plane in the point >S ; and theline X M denote the intersection of the plane E Q e withthe cutting plane, and PM be drawn at right angles to thisline, it can easily be shown that SP : PM :: SA : AX. Hence S and XM represent respectively the other focusand directrix of the ellipse. Also if BCbe the semi-axis minor, and through the centre0 a line UCU be drawn parallel to Ee meeting OD, Odin U and U, then it is evident that BC2 = CU . GU. (3) Let the angle AFO be the angle A EX,,\AEis > AX,.-. AS is > AX,,\ the curve PA is an Hyperbola. Since the angles AFO, FOd are less than the two FOd,FOd, i.e. than two right angles, the lines FA and d 0 maybe produced to meet in A. In this case the cutting plane will intersect the other half ofthe cone, and if any point P be taken on this part of thecurve, and PM be drawn at right angles to XM, it can beshown as before that SP : PM :: SA :: AX. 118 CONIC SECTIONS. D>
Text Appearing After Image:
CONIC SECTIONS. 119 The intersection of the cutting plane therefore with thisportion of the cone will he the other branch of the hyperbola. Also if another sphere be described touching the upperportion of the cone in EQe, and the cutting plane in 8,and the line XM denote the intersection of the planeEQe with the cutting plane, and PM be drawn at rightangles to this line, it can be easily shown that SM : PM :: SA : AX. Hence, S and XM will represent respectively the otherfocus and directrix of the hyperbola. Coe. 1. In this last case, i.e. when the section is an hyper-bola, if a plane OKL be drawn through the vertex of thecone parallel to the cutting plane, meeting the plane of thepaper in the straight line OL, and the surface of the cone in generating line OK; then OL : OK: : OL ::AN: OB,AB, :AX: AE, (Euclid, VI. 2.) : AX : AS, : CA : CS, (Chap. III. Prop. II.) where C is the middle point of A A and therefore the centreof the hyperbola. .. KOL is half the angle between the asymptotes.

Note About Images