Dark shadow

Geometry Level 4

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

  • S1:S_1: Centered at the origin with radius 1.
  • S2:S_2: Centered at (10,2,1)(10,2,1) with radius 3.
  • P:P: Defined by the equation 9x10z=260.9x-10z=260.

The first sphere S1S_1 acts as a light source, the second sphere S2S_2 is opaque, and the plane PP 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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