Show that the equation represents a sphere, and find its center and radius. x^2+y^2+z^2+8 x-6 y+

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

Key Concepts

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions as perfect square trinomials. In the context of equations for spheres, it entails rearranging and regrouping the quadratic and linear terms of each coordinate to facilitate rewriting the equation in a form that clearly displays the center and radius.

Standard Form of a Sphere's Equation

The standard form of a sphere's equation is (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) represents the center and r denotes the radius. Converting a given equation into this form confirms that it represents a sphere and makes it easier to identify its geometric properties.

Center and Radius Determination

Once the equation has been rewritten into the standard form, the coordinates of the center are directly obtained from the terms inside each squared expression, and the radius is the square root of the constant on the right-hand side. This method is essential in analytic geometry for extracting meaningful geometric information from quadratic equations.

Transcript

00:03 Let's look at the equation x squared plus y squared plus z squared equals 8x minus 6y plus 2z plus 17 equals 0, and see if it's the equation for a sphere, and if it is, find it center and radius.

00:19 Well, for this to be an equation for a sphere, we have to be able to write it in this form.

00:25 If we can, then we can read the center from the values h, k, and l, and r squared will be the square of the radius.

00:34 To do this, we're going to have to perform completing the square three times.

00:39 One's for the x part, once for the y part, and once for the z part.

00:44 We'll collect everything on the right -hand side and see if we have a positive value for r -squared.

00:50 That'll tell us that we do have a sphere, and it'll tell us what the radius is.

00:56 First observation to make is that the coefficients of x, squared, y, squared, and z -squared are all the same.

01:02 When we multiply out these binomials, we get x -square -plus -y -6.

01:06 Squared plus z squared.

01:08 All of those have a coefficient of one.

01:10 If we were starting off with an equation where the coefficients of a quadratic terms were not the same, there's no way we could make them all equal to one.

01:23 So to set this up for completing this square, we're going to put the x terms together, the y terms together, the z terms together, put the constants on the right hand side and set things up to perform completing the square on each term.

01:39 So we're going to have x squared plus 8x plus a term to complete the square for the x's plus y squared minus 6y plus a term to complete the square for the y's plus z squared plus 2 z plus a term to be added to complete the square.

02:10 That's going to equal.

02:12 I want to subtract 17 from both sides.

02:15 And then i'm going to leave space for the terms that i have to use to complete the square on the left -hand side, on the right -hand side as well.

02:25 So for the x's, for the y's, and for the zs...

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