Show that the equation represents a sphere, and find its center and radius. x^2+y^2+z^2-2 x-4 y+

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

Key Concepts

Sphere

A sphere is a three-dimensional solid where every point on its surface is equidistant from a fixed central point. Its equation in three-dimensional coordinate space can be written in a standard form which directly provides the center and the radius of the sphere.

Completing the Square

Completing the square is an algebraic technique used to convert a quadratic expression into a perfect square plus or minus a constant. This method is essential when rewriting the general quadratic equation of a sphere into its standard form, thus making it easier to identify the geometric characteristics of the sphere.

Standard Form of a Sphere Equation

The standard form of a sphere's equation is expressed as (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) represents the sphere's center and r is its radius. This form results from reorganizing and completing the square on the general quadratic equation to reveal the sphere's defining elements.

Transcript

00:03 Consider the equation x squared plus y squared plus z squared minus 2x minus 4y plus 8 z equals 15.

00:13 Let's see if this is the equation for a sphere.

00:17 If it is, we should be able to put it into the form of x minus h squared plus y minus k squared plus z minus l squared equals r squared.

00:26 If we can't do that, it's not the equation for a sphere.

00:30 First observation before we do anything else, for there to be any hope, of this being a sphere, the coefficients of x squared, y squared, and z squared all have to be the same.

00:41 Because when we expand this out, we always have x squared plus something, y squared plus something, z squared plus something.

00:49 They'll have to have a coefficient of one.

00:51 And if they don't have a they don't have the same coefficient to begin with, we can't make them all one by doing any type of algebra.

01:01 The key to this is completing the square.

01:04 And so we're going to have to do completing the square three times we're going to have to make the x terms into this form the y terms into this form and the z terms into this form so to set us up for completing the square let's put the x terms together the y terms together and the z terms together and put all of the constants on the other side so i'm going to have x squared minus 2x plus some term that i have to later plus y squared minus 4y plus some term that i'll need to add to complete the square plus z squared plus 8 z plus a term needed to complete the square and that has to be equal to constants 15 that's already there i will have to add the constant that i used to complete the square for x's, i will have to add the constant that i used to complete the square for y's, and a lot of room here, i will have to add the constant that i use to complete the squares for the z's.

02:33 Completing the squared, you take half of the linear coefficient, which we usually call b, and then square it.

02:39 That gives us a perfect square trinomial.

02:42 So if we start with the xes, i have negative 2x...

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