Equation of a circle in 3D space

Hi All,

Thanks for answering my last question about circle in 2D. I got a question about a circle in 3D space here.

There is a sphere at the origin (0,0,0) with a radius of 6. A point at (10, 12, 15) project a cone shape on the sphere where the cone has a cone angle of 10 degree. The concept is similar to the satellite project signal on earth.

So my question is: What is the equation of the circle which is the intersection of the sphere and cone? I know there will be two circles. But I only want the one closer to the point. Thank you!