Given the equations of two spheres, how would I find the equation of any plane tangent to the two spheres?
I tried something, but I realized that it failed, and I am not sure where to go from here. I have only basic knowledge of cross product, dot product, etc. and have not yet taken calculus.
My attempt:
I know the centers of the two spheres. I pick any point on the surface of the first sphere. I find the vector from the center of the first sphere to the point I selected. I then scale the center of the second sphere by the vector I just found divided by the radius of the first sphere and multiplied by the radius of the second sphere. Then, I construct the vector from the point I chose on the first sphere to the point I found on the second sphere. I take the cross product of this vector with the vector formed by the centers of the two spheres. I use this as the normal vector for my plane and plug in to get its equation.
I noticed by experimentation that this does not work. Is there a way of solving this problem in a similar manner to what I tried above?