A Treatise On the Analytic Geometry of Three Dimensions 1

Cover A Treatise On the Analytic Geometry of Three Dimensions 1
A Treatise On the Analytic Geometry of Three Dimensions 1
George Salmon
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The following examples will serve further to illustrate the principles which have been laid down : Ex. 1. To find the locus of the intersection of generaton to a hyperboloid which cat at right angles.
The section parallel to the tangent plane which contains the generaton most be an equilateral hyperbola, so that (Art. 164) (a"«-a'*) + (a"«-o'"»)=0. But (Art. 161) the square of the radius vector to the point is a"s+6"2+c"«-(o"«-a'2)-(a"«-o'"«).
We hare, therefore, the locus a sphere, the square
...of whose radios is equal to a"^-\-V*^+c^\ Otherwise thus: If two generators are at right angles, their plane together with the plane of each and of the normal at the point, are a system of three tangent planes to the surface, mutually at right angles, whose intersection lies on the sphere r«=a"»+6"«+c"« (Art. 93).
Ex. 2. To find the locus of the intersection of three tangent lines to a qoadzio mutually at right angles (see Ex. 6, Art. 121).
Let a, /9, 7 be the angles made by one of these tangents with the normals thioogh the locus point, and since each of these tangents lies in the tangent cone thxou^ that point, we hare the conditions cos'g cos*^ ootf y _ oos'g' c oe*/y oos'y^ _ »'«-.«« a"^-'a^ a'"*-a^ ~ ' coB» a" cos*jy' COS* y" __ FOCAL CONICS AND CX)NFOCAL SURFACES.


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