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Part of 2017 CMIMC Geometry

Problems(1)

2017 G4: Hey Look, Coordinates!

Source:

1/29/2017
Let S\mathcal S be the sphere with center (0,0,1)(0,0,1) and radius 11 in R3\mathbb R^3. A plane P\mathcal P is tangent to S\mathcal S at the point (x0,y0,z0)(x_0,y_0,z_0), where x0x_0, y0y_0, and z0z_0 are all positive. Suppose the intersection of plane P\mathcal P with the xyxy-plane is the line with equation 2x+y=102x+y=10 in xyxy-space. What is z0z_0?
2017geometryanalytic geometry