Show that the equation represents a sphere, and find its center and radius. 2 x^2+2 y^2+2 z^2=8

Show that the equation represents a sphere, and find its...

Key Concepts

Sphere in Analytic Geometry

A sphere in three-dimensional space is defined as the set of all points that are at a constant distance (the radius) from a fixed point (the center). Understanding spheres in this context is crucial for analyzing and representing three-dimensional geometry and for solving equations that describe spherical surfaces.

Completing the Square

Completing the square is an algebraic technique used to transform quadratic expressions into a perfect square plus a constant. This method is essential for rewriting equations in standard forms, which makes it easier to identify important properties like centers and radii in the context of circles and spheres.

Standard Form of the Sphere Equation

The standard form of a sphere's equation is (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) is the center and r is the radius. Converting a general quadratic equation to this form helps in clearly identifying the geometrical attributes of the sphere.

Extraction of Geometric Properties

By rewriting the equation of a sphere in its standard form through techniques such as completing the square, one can directly extract the geometric properties of the sphere, including its center and radius. This approach is fundamental in linking algebraic manipulation with geometric interpretation.

Transcript

00:02 Let's look at the equation 2x squared plus 2y squared plus 2z squared equals 8x minus 24z plus 1 and see if this represents the equation of a sphere.

00:16 If it does, we should be able to write it in this form, where h, k, and l will be the center of our sphere.

00:25 And the right -hand side is going to be a positive number to represent the radius squared.

00:29 To do that, we'll involve a little bit of completing the square plus some other algebra.

00:36 That may be needed.

00:38 First thing to notice is that the coefficients of x squared, y squared, and z squared are all the same.

00:45 That's going to allow me to divide them each by the same number to make their coefficients one, because the coefficients of x squared, y squared, and z squared when we expand out this form of the equation are all going to be one.

00:59 So let me do that first.

01:01 Let me take this equation and multiply both sides of this equation by one -half, which is mathematically the same thing as dividing by two, and that will make all of the quadratic coefficients one, which is what this form of the equation needs.

01:22 Now remember the one -half is going to distribute, and multiply it by the 2x squared, the 2 -y -squared, and the 2 -z -squared, dividing each of those terms by two, or canceling out their coefficients.

01:36 And so that's going to give me x squared plus y squared plus z squared equals and on the right hand side it's going to distribute as well giving me 4x minus 12 z plus one half so that's the first step is to make our coefficients of x squared y squared and z squared equal to one now the next thing i want to do is i want to put all of the variable terms on the left -hand side, put all constant terms on the right -hand side, and that will set me up for the completing the square step that i will have to do on at least two of these particular terms.

02:29 So i'm going to have x squared, subtracting 4x from both sides, is going to give me minus 4x, plus a term to be later, to be named later.

02:43 Then there are no terms to go with the y squared.

02:47 There's no y term, and so we're not going to have to add any additional terms.

02:53 It's already a perfect square by itself.

02:56 And then for the z, we have z squared.

03:03 Let's add 12 z to both sides.

03:05 That gives me 12 z plus a term to be named later.

03:12 That's going to be equal to the one -half that's already there.

03:16 The term we add to complete the square for x's, plus the term we add to complete the square for y's, and the 4x and the negative 12z were canceled by the minus 4x and the plus 12z that we added to both sides.

03:38 So completing the square for x, half of negative 4 is negative 2.

03:46 So i'm going to have to take negative 2 squared and add it to both sides...

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