Answer
$(x,y,z) = (-3,-3,-6)$
$(x,y,z) = (3,3,6)$
Work Step by Step
$(x,y,z) = (0,0,0) + n(1,1,2) = (n, n, 2n)$
We can find the two values of $n$ such that the points on the line satisfy the equation of the sphere:
$x^2+y^2+z^2 = 54$
$(n)^2+(n)^2+(2n)^2 = 54$
$ 6n^2 = 54$
$n^2 = 9$
$n^2-9 = 0$
$(n+3)(n-3) = 0$
$n=-3~~$ or $~~n=3$
We can find the two points:
$(x,y,z) = (0,0,0) + (-3)(1,1,2) = (-3,-3,-6)$
$(x,y,z) = (0,0,0) + (3)(1,1,2) = (3,3,6)$
