Question

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!

Answer 1

if the given points are on the surface of a sphere then the distance from the center of the sphere to each point is the same... equal to the radius of the sphere.

let the center of the sphere be (x,y,z)
then you know (by the distance formula between 2 points)
(x-0)^2 + (y-0)^2 + (z-0)^2 = r^2
(x-12)^2 + y^2 + z^2 = r^2
etc.
you can solve for x,y and z

Answer 2

Can you elaborate? I know all those points are on the sphere. So i substitute some of them in. Do i use simultaneous equations then?

Answer 3

yes, you solve simultaneous equations

for example, from the first two equations (of 4 you can make)
x^2 + y^2 + z^2 = r^2
x^2 -24x + 144 + y^2 + z^2 = r^2

subtract to get
24x -144 =0
24x= 144
x = 6

Answer 4

Ok i got the sphere equation of (x-6)^2 + (y-3)^2 + (z-2)^2 = 49
How do i show that the center (6,3,2) is outside of the pyramid?
@Hero do you have an idea about this?

Answer 5

if you plot the center point of the circle and compare it to the pyramid, you can see the point is outside of the pyramid.

If you must do it analytically, each side of the pyramid is contained in a different plane.
In particular, the "slanted" plane that contains the points
(12,0,0), (0,6,0) and (0,0,4) has the equation
x+2y+3z= 12
if you substitute in the point (6,3,2) you get
6+6+6 = 18
Because 18 is bigger than 12, this shows that the center point is "on the other side" of the plane from the origin (which gives 0)
and that means (6,3,2) is on the wrong side to be inside the pyramid.

Answer 6

That almost is very clear, but can you please explain the part with the 12 and 18? I know how to find that plane, and sort of substitute in the values, and understand how you get the 18.

I just don't quite get how that represents the distance... Can you make a simple sketch with those values?

Answer 7

Here is one way to think of it.

Imagine you draw a vector (line) from the origin and going in some direction


and go "d" steps along the line.