In each part, find the standard equation of the sphere that...

In each part, find the standard equation of the sphere that satisfies the stated conditions.
(a) Center (1,0,-1)$;$ diameter $=8$
(b) Center (-1,3,2) and passing through the origin.
(c) A diameter has endpoints (-1,2,1) and (0,2,3)

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Key Concepts

Midpoint Formula

The midpoint formula in three dimensions, which calculates the midpoint of a segment connecting two points as ((x? + x?)/2, (y? + y?)/2, (z? + z?)/2), is vital when the endpoints of a diameter are provided. The midpoint represents the center of the sphere.

Distance Formula in Three Dimensions

The distance formula, given by d = ?[(x? - x?)² + (y? - y?)² + (z? - z?)²], is used to compute the distance between any two points in three-dimensional space. This formula is essential for determining the radius when a sphere's center and another point on its surface are known.

Standard Equation of a Sphere

The standard equation of a sphere in three-dimensional space is given by (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) represents the coordinates of the center and r is the radius of the sphere. This form is fundamental in analytic geometry for representing all points that are equidistant from a fixed center.

Relationship Between Diameter and Radius

The radius of a sphere is half the length of its diameter. This relationship is crucial when only the diameter is given, as it allows for the straightforward calculation of the radius needed for the sphere’s equation.

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