Dark shadow

Geometry Level 3

In a Cartesian coordinate system, there are 2 spheres and 1 plane described as follows:

  • \(S_1:\) Centered at the origin with radius 1.
  • \(S_2:\) Centered at \((10,2,1)\) with radius 3.
  • \(P:\) Defined by the equation \(9x-10z=260.\)

The first sphere \(S_1\) acts as a light source, the second sphere \(S_2\) is opaque, and the plane \(P\) is a screen.

What is the area of the umbra (the darkest part of the shadow, where an observer cannot see the light source at all) on the screen?

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