Identifying a sphere
You are given three points in the Cartesian 3D space: , and you are asked to construct a sphere that passes through , and at the same time be tangent to the plane. There are two spheres that satisfy these requirements. Select the smaller one and let its radius be . Enter as your answer.
The answer is 63751.
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1 solution
The center of the sphere is at ( x 0 , y 0 , R ) . All three of these numbers are unknown. Solve the following system of nonlinear equations for these values. Newton Raphson works well here.
( A x − x 0 ) 2 + ( A y − y 0 ) 2 + ( A z − R ) 2 = R 2 ( B x − x 0 ) 2 + ( B y − y 0 ) 2 + ( B z − R ) 2 = R 2 ( C x − x 0 ) 2 + ( C y − y 0 ) 2 + ( C z − R ) 2 = R 2
Solution 1:
x 0 = 1 . 9 4 9 9 y 0 = 5 . 5 8 5 6 R = 6 . 3 7 5 1
Solution 2:
x 0 = − 1 3 . 9 4 9 9 y 0 = − 1 2 . 5 8 5 6 R = 4 6 . 1 2 4 8