An elementary treatise on coordinate geometry of three dimensions . hree conjugate diameters of an ellipsoid meet thedirector sphere in P, Q, R, the plane PQR touches the ellipsoid. Ex. 10. Find the equation to the normal plane ( at right anglesto the tangent plane) of the cone ax2 + bi/- + cz2 = 0 which passes throughthe generator xjl =yjm=z/n. , v(6—c)ar n A H$. ~i j — U. 122 COORDINATE GEOMETRY [CH. VII. Ex. 11. Lines drawn through the origin at right angles to normalplanes of the cone ax2 + by2+cz2 = 0 generate the cone a(b — c)2 b(c-a)2 c(a-b)2 9 1 o \-- s—=U. or yz 2 Ex. 12. If the t
An elementary treatise on coordinate geometry of three dimensions . hree conjugate diameters of an ellipsoid meet thedirector sphere in P, Q, R, the plane PQR touches the ellipsoid. Ex. 10. Find the equation to the normal plane ( at right anglesto the tangent plane) of the cone ax2 + bi/- + cz2 = 0 which passes throughthe generator xjl =yjm=z/n. , v(6—c)ar n A H$. ~i j — U. 122 COORDINATE GEOMETRY [CH. VII. Ex. 11. Lines drawn through the origin at right angles to normalplanes of the cone ax2 + by2+cz2 = 0 generate the cone a(b — c)2 b(c-a)2 c(a-b)2 9 1 o \-- s—=U. or yz 2 Ex. 12. If the two cones ax2 + by2 + cz2=0, + /3y2 + yz2=0 haveeach sets of three mutually perpendicular generators, any two planeswhich pass through their four common generators are at right angled. THE The locus of the equation x2 . y _2za b2 W ^,2 7,2 ~ - x2 y2_2z The equation (1) represents the surface generated by the y 2k variable ellipse z = k, -% + — =— This ellipse is imaginaryunless Jc and c have the same sign, hence the centre of the z. Fig. 38. ellipse lies on OZ if c>0 and on OZ if c z—k. The hyperbola Is real for all real values of k, and its centre passes in turnthrough every point on zz. When /; = () the hyperbola degenerates into the two lines ~ — L = 0, 0=0. The sections of the surface by the planes z = k, z=—k project on the
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
