# 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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#### Solution Preview

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.

Euclidean three space

Suppose instead of working on the Euclidean plane we study geometry on a sphere in (Euclidean) three space. We interpret point to mean any point on the sphere and we interpret line to mean any great circle on the sphere (that is any circumference of the sphere).

a. Is Euclidâ??s parallel postulate true in this setting?

b. How would you define the angle which two great circles make at a point where they

intersect on the sphere?

c. How would you define a triangle on the sphere? Give some examples.

d. Is the sum of the angles in a triangle greater than, less than, or equal to 180 degrees in this setting?

e. Does the angle sum depend on the area of the triangle? How?

f. How does part d relate to showing that the parallel postulate implies the triangle postulate?