An elementary treatise on coordinate geometry of three dimensions . oint on it. Then the coordinates of Q are xQ, y0. :„.and since P is on the curve, f(x0, y0) = 0, thus the coordinatesof Q satisfy the equation f(x,y) = i). Therefore the co-ordinates of every point on PQ satisfy the equation andevery point on PQ lies on the locus of the equation. ButP is any point of the curve, therefore the locus of theequation is the cylinder generated by straight lines drawn 10 COOEDINATE GEOMETRY fCH. I. parallel to OZ through points of the curve. Similarly,f(y, z) = 0, f(z, x) = 0 represent cylinders gene
An elementary treatise on coordinate geometry of three dimensions . oint on it. Then the coordinates of Q are xQ, y0. :„.and since P is on the curve, f(x0, y0) = 0, thus the coordinatesof Q satisfy the equation f(x,y) = i). Therefore the co-ordinates of every point on PQ satisfy the equation andevery point on PQ lies on the locus of the equation. ButP is any point of the curve, therefore the locus of theequation is the cylinder generated by straight lines drawn 10 COOEDINATE GEOMETRY fCH. I. parallel to OZ through points of the curve. Similarly,f(y, z) = 0, f(z, x) = 0 represent cylinders generated byparallels to OX and OY respectively. Ex. What surfaces are represented by (i) x2+y2 = a2, (ii) y2 = 4ax,the axes being rectangular ? Two equations are necessary to determine the curve inthe plane XOY. The curve is on the cylinder whose equa-tion is f(x, y) = 0 and on the plane whose equation is z = 0,and hence the equations to the curve are f(x, y) = Q,z = 0. Ex. What curves are represented by(i) x2+y2=a2, 2=0 ; (ii) x2+y2=di, z—b ; (iii) z2=4ax, y=o%. (The surface shewn is represented by the equationa2xi + biyi—zi.) Consider now the equation fix, y, z) = 0. The equationz = k represents a plane parallel to XOY, and the equationf(x, y, Z?) = 0 represents, as we have just proved, a cylinder §9] THE EQUATION TO A 81 ll \/, z) = (), and hence/(./?, y, k) = 0 represents the cylinder generated by linecparallel to OZ which pass through tin- common points, (fi^r. 6 iThe two equations f(x, y, k) = 0, z = k represent the curveof section oi* the cylinder by the plane z = J>; which is thecurve oi section of the locus by the plane Z—Tc If, real values from — cc to +x be given to k, the curvef(x, y, k) = 0, z = k, varies .continuously and generate Bsurface. The coordinates of every point on this
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
