An elementary treatise on coordinate geometry of three dimensions . revolu-tion of the circle x2+y2 + 2ax + b2 = 0, z — 0, about the (x2+y2 + z2 + b2)2 = 4a2(x2 + z2). Ex. 6. Sketch the forms of the surfaces : (i) {y2 + z2)(2a — x)=x3, (ii) r2 = a2 cos 20, (iii) u2=2ez. The surfaces are generated by rotating (i) the curve y2(2a-x)=x3about OX ; (ii), the lemniscate in the plane ZOX, r2 = a2cos20, aboutOZ ; (iii) the parabola in the plane YOZ, y2 = 2cz, about OZ. Ex. 7. Prove that the locus of a point, the sum of whose distances from the points (a, 0, 0), (- a, 0, 0) is constant, (2k
An elementary treatise on coordinate geometry of three dimensions . revolu-tion of the circle x2+y2 + 2ax + b2 = 0, z — 0, about the (x2+y2 + z2 + b2)2 = 4a2(x2 + z2). Ex. 6. Sketch the forms of the surfaces : (i) {y2 + z2)(2a — x)=x3, (ii) r2 = a2 cos 20, (iii) u2=2ez. The surfaces are generated by rotating (i) the curve y2(2a-x)=x3about OX ; (ii), the lemniscate in the plane ZOX, r2 = a2cos20, aboutOZ ; (iii) the parabola in the plane YOZ, y2 = 2cz, about OZ. Ex. 7. Prove that the locus of a point, the sum of whose distances from the points (a, 0, 0), (- a, 0, 0) is constant, (2k), is the ellipsoid X V ~f~ zuof revolution -rs+ja 2 = 1- SSii-WJ CHAPTER 12. The angle that a given directed line OP makes witha second directed line OX we shall take to be the smallestangle generated by a variable radius turning in the planeXOP from the position OX to the position OP. The sign ofthe angle is determined by the usual convention. Thus, infigures 9 and 10, 0X is the positive angle, and the negativeangle that OP makes with Fio. 0. Fio. 10. 13. Projection of a segment. If ab is a given segrru ntand A, B are the feet of the perpendiculars from A, B to agiven line Xx, the segment ab is the projection of flicsegment AB on xx. From the definition it follows that the projection of BAis BA, and therefore that the projections of AB and BAdiffer only in sign. It is evident that AB is the intercept made on XX bythe planes through A and B normal to XX, and hence theprojections of equivalent segments are equivalent segments. 14. If AB is a given segment of a directed line MNwhose positive direction, MN, makes an angle 0 with a 16 COORDINATE GEOMETRY [CH. II. given line XX, the projection of AB on XX is equal toAB. cos 0. In figures 11 and 12, AB is positive, in figures 13 and 14,AB is negative.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
