Here are the givens: Triangular pyramid with coordinates of (0,0,0) origin A(12,0,0) B(0,6,0) and C(0,0,4). Determine the equation of a sphere that passes through the vertices of the pyramid OABC.
Part 2: Then once I have the equation of this sphere, how do i show that the center of the sphere DOES NOT lie inside of the pyramid? I don't know the method with checking if its on the same side of all planes or whatnot, I'm still in highschool, so I need the most simple mathematical way to show that. Thanks a lot in advance!
The center of the sphere (cyan) is the red ball with center (6,3,2) the sphere has radius 7 . Its equation is:
$$(x - 6)^2 + (y - 3)^2 + (z-2)^2 = 49$$
The coordinate planes are in grey , $ x=0$, $y=0$ , and $z=0$ , plus one more plane ( shown in pink) it has equation: $ z = -x/3 - 2y/3 + 4 $ . The point at (0,0,0) can't be seen in the picture (which is great because we know it is the origin of the coordinate system and the pink plane prevents us from viewing it.)
So ,what is a natural way to show that (0,0,0) and (6,3,2) are separated from each other by the plane $ z = -x/3 - 2y/3 + 4 $ ?