An elementary treatise on coordinate geometry of three dimensions . ion S + Aw2=0. Ex. 7. Prove that if a straight line has three points on a conicoid,it lies wholly on the conicoid. (The equation (1), § 66, is an identity.) Ex. 8. A conicoid passes through a given point A and touches agiven conicoid S at all points of the conic in which it is met by thepolar plane of A. Prove that all the tangents from A to S lie on find the equation to the enveloping cone of S whose vertex is A. Ex. 9. The section of the enveloping cone of the ellipsoid%2/a2+y2Jb2 + z2lc2 = l whose vertex is P by th
An elementary treatise on coordinate geometry of three dimensions . ion S + Aw2=0. Ex. 7. Prove that if a straight line has three points on a conicoid,it lies wholly on the conicoid. (The equation (1), § 66, is an identity.) Ex. 8. A conicoid passes through a given point A and touches agiven conicoid S at all points of the conic in which it is met by thepolar plane of A. Prove that all the tangents from A to S lie on find the equation to the enveloping cone of S whose vertex is A. Ex. 9. The section of the enveloping cone of the ellipsoid%2/a2+y2Jb2 + z2lc2 = l whose vertex is P by the plane z = 0 is (i) aparabola, (ii) a rectangular hyperbola. Find the locus of P. ,.., .v2+v2 z2 ,Arts. (i)z=±c, (n)^f2 + ^L 74. The locus of the tangents which are parallel toa given line. Suppose that PQ is any chord and that M isits mid-point. Then if the line PQ moves parallel to itselftill it meets the surface in coincident points, it becomes atangent and M coincides with the point of contact. There-fore the point of contact of a tangent which is parallel. Fig. 35. to a given line lies on the diametral plane which bisects allchords parallel to the line. This plane cuts the surface ina conic, and the locus of the tangents parallel to the givenline is therefore the cylinder generated by the parallels tothe given line which pass through the conic. g§74,75] THE ENVELOPING CYLINDER ill Let (a, f-i, y) (fig. 35) be any poinl on a tangeni parallelfco a given line x I = y/m — since, by $7:; 11), the line x — — f-i_z — yI m n touches the .surface if {al2+bm2 + cn2)( + b/32+ cy2-\) = (<;d + bfim + cyaf,the locus of (a, $, y) is given by(al2+bm2 + en 2){ax2 + by2 + cz2 -1) = (ate + &m 2/ + enzf. This equation therefore represents the enveloping cylinder,which is the locus of the tangents. The enveloping cylinder may be considered to be a limiting case ofthe enveloping cone whose vertex is the point P, (Ir, mr, nr) on theline .v/l=y/m = z/n, as r
Size: 1337px × 1869px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
