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Equation of a Sphere and Geometric Plane

I have the solution to these problems. Solution is included but need the calculations to find these solutions. Please see attached document.

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4. a) The equation for a sphere of radius R centered at (x0, y0, z0) is as follows:

(x - x0)^2 + (y - y0)^2 + (z - z0)^2 = R^2.

Since the center passes through the plane x + y + z = 0, we have x0 + y0 + z0 = 0.

Plugging in the points (1, -3, 4), (1, -5, 2), (1, -3, 0) contained in the sphere, we obtain the following three equations:

(1) (1 - x0)^2 + (-3 - y0)^2 + (4 - z0)^2 = R^2
(2) (1 - x0)^2 + (-5 - y0)^2 + (2 - z0)^2 = R^2
(3) (1 - x0)^2 + (-3 - y0)^2 + (0 - z0)^2 = R^2

Expanding the squares, we obtain the following:

(1') x0^2 - 2 x0 + 1 + y0^2 + 6 y0 + 9 + z0^2 - 8 z0 + 16 = R^2
(2') x0^2 - 2 x0 + 1 + y0^2 + 10 y0 + 25 + z0^2 - 4 z0 + 4 = R^2
(3') x0^2 - 2 x0 + 1 + y0^2 + 6 y0 + 9 + z0^2 = R^2

Subtracting (1') from (3'), we obtain 8 z0 - 16 = 0, which implies z0 = 2.

Subtracting (3') from (2'), we obtain

4 y0 + 16 - 4 z0 + 4 = R^2.

Plugging in z0 = 2, this becomes

4 y0 + 16 - 8 ...

Solution Summary

We solve three problems in analytic geometry pertaining to spheres, planes, and ellipsoids.

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