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:
$(x,y,z) = (3,4,5) + (1)(3,4,-5) = (6, 8, 0)$
$(x,y,z) = (3,4,5) + (-1)(3,4,-5) = (0, 0, 10)$
