An elementary treatise on coordinate geometry of three dimensions . ves, and certain sets of three of the six planes ofintersection, one from each pair of conicoids, pass through one line. Ex. 6. Prove that the ellipsoid and sphere given by . B2 + 6ya + 14;- = 200, 5 (.<•- + //- + :2) - + 36- + 20 = 0have double contact, and that the chord of contact is .r = 8, 1=2. Ex. 7. If a system of conicoids has a common conic section, thepolar planes with respect to them of any point in the plane of thesection pass through one line. 250 COORDINATE GEOMETRY [ch. xii. Ex. 8. If two conicoids have
An elementary treatise on coordinate geometry of three dimensions . ves, and certain sets of three of the six planes ofintersection, one from each pair of conicoids, pass through one line. Ex. 6. Prove that the ellipsoid and sphere given by . B2 + 6ya + 14;- = 200, 5 (.<•- + //- + :2) - + 36- + 20 = 0have double contact, and that the chord of contact is .r = 8, 1=2. Ex. 7. If a system of conicoids has a common conic section, thepolar planes with respect to them of any point in the plane of thesection pass through one line. 250 COORDINATE GEOMETRY [ch. xii. Ex. 8. If two conicoids have two common generators of the samesystem, they have two other common generators. Ex. 9. The centres of conicoids which have double contact withthe surface ax*+bf+cz*=\ at its points of intersection with the chord x=cl, y = /3, and intersectthe plane z=0 in a circle, lie on the line ,=0 JL_JL=l_l. aa. bfi a b Ex. 10. A sphere of constant radius r has double contact with the ellipsoid 2 2 ,2 ^r+- + - = b2 c2 Prove that its centre must lie on one of the conies. Examine when the contact is real arid when the sphere lies whollywithin the ellipsoid. Cf. § 130. Ex. 11. If a conicoid is circumscribed to a sphere, every tangentplane to the sphere cuts the conicoid in a conic which has a focus atthe point of contact. Ex. 12. If a conicoid is circumscribed to another conicoid, thetangent plane to either at an umbilic cuts the other in a conic ofwhich the umbilic is a focus. Ex. 13. Any two enveloping cones of the conicoid ax2 + by2 + cz2—l whose vertices lie on the concentric and nomothetic conicoid ax2 + by2 + cz2 = k2have double contact. Ex. 14. The centres of conicoids which have double contact witha given conicoid so that the chord of contact is parallel to a given linelie in a given plane. Ex. 15. If two cones have a common circular section, they havedouble contact, and if the line joining their vertices meet the plane ofthe circle in P, the chord of contact is the polar of
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
