Identify the graph of the equation obtained by substituting...

Identify the graph of the equation obtained by substituting the coordinates of the given points in the equation of Exercise 34. If the graph is a sphere, find its radius and the coordinates of its center.
$\mathbf{Q}(0,0,0), \mathbf{R}(0,1,0), \mathbf{S}(4,0,2), \mathbf{T}(-1,2,3)$

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Key Concepts

Completing the Square

Completing the square is a method used to rearrange quadratic equations into standard form. In the context of a sphere's equation, it allows one to rewrite the equation so that the center (a, b, c) and the radius r become apparent, which is crucial for graphing and identifying the geometric properties of the sphere.

Substitution of Coordinates

When determining the equation of a sphere through specific points, each point’s coordinates are substituted into the general sphere equation. This process creates a system of equations that, when solved, reveal the values of the unknown parameters, such as the center coordinates and the radius.

Equation of a Sphere

In three-dimensional analytic geometry, a sphere is defined as the set of all points that are at a fixed distance (the radius) from a given point (the center). Its standard equation is written as (x - a)² + (y - b)² + (z - c)² = r², where (a, b, c) represents the center and r is the radius.

Solving Systems of Equations

The substitution of multiple points into the sphere’s equation leads to a system of equations. Techniques such as elimination or substitution are used to solve this system, which is essential for finding the sphere's center and radius.

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