Welcome to a lesson on the Equation of a sphere.
The goal of the video is to write an equation of a sphere in standard form,
to identify the center, and the radius of the sphere.
Here’s the graph of a sphere, as you can see it looks like a ball.
The definition of a sphere
is the set of points in R3, or 3 Dimensions equal distance from a given point, or the center.
Looking at this sphere, the center would be the origin.
We’ll call this the point (a,b,c) in the most general sense.
And the radius would be the distance from the center to the surface of the sphere.
So if we have the equation of a sphere in standard form, we can identify both the center and the radius.
So as we just stated, a sphere is a set of points in space equal distance from a point called the center.
If a sphere has a center (a,b,c) with radius r,
the equation of the sphere is given by:
the quantity x minus a squared, plus the quantity y minus b squared, plus the quantity z minus c squared, equals r squared.
So again in this form we can identify the center as well as the radius.
Let’s take a look at a couple of example problems.
Here we want to determine the center and radius of the given sphere.
So we have to re-write this equation in standard form as we see here below,
to identify both the center and the radius.
So what we’re gonna do first is group the x terms together, the y terms together, and the z terms together.
So we’ll have x squared minus 2x,
and we’ll leave some space and put plus y squared,
plus 6y, leave a little bit of space and put plus z squared plus 8z.
We’re also gonna go ahead and subtract 1 on both sides.
Because you know we have to have our constant term on the right side of the equation.
Now we’ll go ahead and complete the square and on the x, y, and z part of the equation,
so that we can create perfect square trinomials for each variable.
Remember, to find the constant to make a trinomial a perfect square, we have to take half of b,
and then square it.
We’ll half of negative 2 would be negative 1, and negative 1 squared would be positive 1.
So if we add 1 right here, this would now be a perfect square trinomial.
But if we add 1 to the left side of the equation, we also have to add 1 to the right side of the equation.
So we’ll put a plus 1 here to maintain the equality.
Now let’s go ahead and take a look at the y part,
if we take half of 6 take would be 3 and then square it, we have to add 9.
And we can do that as long as we also add 9 to the right side.
Now this should be a perfect square.
Now we'll go ahead and take a look at the z part. Half of 8 would be 4,
and 4 squared is 16. So we'll had 16 here, and then we'll also add 16 on the right side.
Now let’s go ahead and factor the x, y, and z parts on the left side.
The factors of positive 1 that add to negative 2 are negative 1 and negative 1.
So we'd have x minus 1 times x minus 1. Or x minus 1 squared.
Plus, the factors of 9 that add to 6 would be positive 3 and positive 3.
So we’d have y plus 3 times y plus 3, or y plus 3 squared.
And then for the z part, the factors of 16 that add to 8 would positive 4 and positive 4.
So instead of writing z plus 4 and z plus 4, write z plus 4 squared.
And on the right side we have negative 1 plus 1 , that's of course 0.
And then we have 9 plus 16 which equals 25.
So this is now in the form where we can identify the center and the radius.
Notice down here each binomial is x minus, y minus, or z minus.
So we have to pay close attention to the sign of the coordinates of this center.
If it’s x minus 1, then the center would have an x coordinate of positive one.
If its y plus 3,
the y coordinate would be negative 3,
and if we have z plus 4, the z coordinate would be negative 4.
I always just think it's always the opposite sign of what we see in order to fit this special form.
But really, if this had to be minus, it would have to be y minus a negative 3,
and that’s the reason why this coordinate is negative 3.
Now to find the radius, we know that r squared is equal to 25.
We take the square root of both sides we have r equals plus or minus 5,
but since we know the radius has to be a length,
we''ll only use the positive value. So r or the radius is equal to 5 units.
Let’s go ahead and take a look at one more.
We want to find the center and the radius of the given sphere again, same question but now notice we have coefficients
on the x squared, y squared, and z squared terms.
Let’s go ahead and group the x’s, y’s, and z’s together.
So we’ll have 4x squared minus 8x
Leave some space.
Plus 4y squared plus 16y,
and we have plus 4 z squared, and notice there’s no z term.
And that’s okay.
Let’s also go ahead and add 1 to both sides of the equation so we have 1 on the right side.
Now in order to complete the square, we have to have a leading coefficient of 1.
So what we have to do is factor a 4 out of all these squared terms.
Let’s do that next.
If we factor out 4 we’d be left with x squared minus 2x,
and we are gonna complete the square on this part so I’ll leave some space.
Plus, factor out a 4 from the y and y squared terms.
We’d have 4 times y squared plus 4 y.
And we’ll leave some space.
We’ll go and factor out the 4 from here. So we’ll have 4 times z squared...equals 1.
Now we’ll go and complete the square for the x part and the y part.
So if we take half of negative 2 and square it we’ll have negative 1 squared or positive 1.
And here’s where we have to be careful if we put a positive 1 here,
if we distribute this 4 we’re actually adding 4 by putting this plus 1 here.
So we have to add 4 to the right side of the equation.
Let’s take a look at the y part now.
Take half of 4, that would be 2 and square it that would be positive 4.
But putting a plus 4 here,
we are making this a perfect square but we’re actually adding 16 if we distribute we’d have 4 times 4.
So we have to add 16 to the right side as well.
And then for the z part we don’t have to complete the square because there’s no z term.
Let’s go ahead and factor these.
We have 4 times; this is gonna end up being x minus 1 squared.
The factors of positive 1 that add to negative 2 are negative 1 and negative 1.
And instead of writing x minus 1 times x minus 1, we can write the quantity x minus 1 squared.
Plus, 4 times... the factors of 4 that add to 4 would be 2 and 2, so we end up having y plus 2 squared.
Plus 4 times,
now to fit this form perfectly im gonna go ahead and rewrite this as the quantity z minus 0 squared.
Of course subtracting 0 doesn’t change z, this is still z squared.
But it does fit this form perfectly.
And then on the right side were gonna have 1 plus 4 plus 16, that’s 21.
What we should notice now is there's this extra factor of 4.
Here, here, and here. Which is not in the form we want.
What were gonna do now is divide everything by 4.
Divide this by 4, this by 4, this by 4, and also the right side by 4.
Now let’s go ahead and simplify this.
And notice we have 4 over 4, 4 over 4, and 4 over 4. All of this simplifies nicely,
and we’re left with the quantify x minus 1 squared.
Plus, quantity y plus 2 squared. Plus the quantity z minus 0 squared equals 21/4
Okay so from this form we can identify the center and the radius.
The center would be positive 1 negative 2. And the z coordinate would be 0.
And r squared is equal to 21 divided by 4.
So r is going to equal the square root of 21 divided by 4.
This does simplify.
The numerator would be the square root of 21,
denominator would be the square root of 4 which is equal to positive 2.
And again, remember the radius is a length so we only use the positive value for r.
So here's our center, and here's our radius.
Okay I hope you found these examples helpful. Thank you for watching.