Question

Find the points on the sphere x² +y² +z²= 36 that are closest to and farthest from the point (1,2,2).

Answer 1

let's see .. let x1,y1,z1 be the points on sphere

for the points to be closest, 1,2,2 must line in the radius connecting x1,y1,z1

so, x1/sqrt(x1^2+y1^2+z1^2) = 1/sqrt(1^2+2^2+2^2)
y1/sqrt(x1^2+y1^2+z1^2) = 2/sqrt(1^2+2^2+2^2)
z1/sqrt(x1^2+y1^2+z1^2) = 2/sqrt(1^2+2^2+2^2)
as we know, r = 6, we have sqrt(x1^2+y1^2+z1^2) = 6

x/6 = 1/sqrt(1^2+2^2+2^2);
y/6 = 2/sqrt(1^2+2^2+2^2);
z/6 = 2/sqrt(1^2+2^2+2^2)

must give the closest point. so your answer must be
http://www.wolframalpha.com/input/?i=x%2F6+%3D+1%2Fsqrt%281%5E2%2B2%5E2%2B2%5E2%29%3B+y%2F6+%3D+2%2Fsqrt%281%5E2%2B2%5E2%2B2%5E2%29%3B+z%2F6+%3D+2%2Fsqrt%281%5E2%2B2%5E2%2B2%5E2%29

for the farthest point, we have the other point on the other side of the sphere
(-x,-y,-z) for the same values