Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10:
ISBN 13:

Chapter 10 - Section 10.6 - The Three-Dimensional Coordinate System - Exercises - Page 482: 37

Answer

$(x,y,z) = (6, 8, 0)$ $(x,y,z) = (0, 0, 10)$

Work Step by Step

$(x,y,z) = (3,4,5) + n(3,4,-5) = (3+3n, 4+4n, 5-5n)$ We can find the two values of $n$ such that the points on the line satisfy the equation of the sphere: $x^2+y^2+z^2 = 100$ $(3+3n)^2+(4+4n)^2+(5-5n)^2 = 100$ $(9+18n+9n^2)+(16+32n+16n^2)+(25-50n+25n^2) = 100$ $50n^2+50 = 100$ $50n^2-50 = 0$ $50(n^2-1) = 0$ $50(n+1)(n-1) = 0$ $n=-1~~$ or $~~n=1$ We can find the two points: $(x,y,z) = (3,4,5) + (1)(3,4,-5) = (6, 8, 0)$ $(x,y,z) = (3,4,5) + (-1)(3,4,-5) = (0, 0, 10)$
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