Circles: Equations, Centers, and Radii
October 26, 2026

[00:00:05] Okay. If you use the square root principle, you get y is plus or
[00:00:15] minus the square root of negative x squared plus nine, which is this and this. These two are functions.
[00:00:37] This is not a function. Is this a quadratic function?
[00:00:46] Is the square root of x a polynomial? That same thing is x to what power? One half power. Is that a polynomial?
[00:01:01] The exponents of the variables have to be whole numbers for it to be a polynomial. These square-root functions are not polynomials.
[00:01:13] So therefore these are functions, but they are not polynomial functions.
[00:01:24] So this is not a function. We could break it up into two separate functions, but it is a relation.
[00:01:29] Okay. And it has lots of solutions. Y squared negative x squared plus nine. Let's and we often actually write like this.
[00:01:46] X squared plus y squared equals nine. We'll come back to that. What is this gonna look like? Let's take a look.
[00:02:02] It turns out x squared plus y squared equals nine is a circle. And what's the radius of the circle?
[00:02:17] Three. So why in the world does this give us a circle? I'll kind of give you an idea. If I take the square root of both sides.
[00:02:36] Does this thing look familiar from last year or the year before?
[00:02:45] What if I ask you to find the distance between zero, zero, and x, y? What's the distance formula?
[00:03:03] If we have two ordered pairs, this is the change in x. This is the change in y. Okay. So the distance between them is basically the Pythagorean Theorem.
[00:03:18] The difference of the x's, the difference of the y's squared plus squared is the Pythagorean Theorem, which gives us this. Anyways, long story short, this is the distance formula between these two ordered pairs, zero, zero, and the ordered pair x, y.
[00:03:47] So everything is a solution to this. If it's the distance from zero, zero, the distance is equal to three. In fact, that is what a circle is. The set of ordered pairs is the same distance away. In this case, zero, zero.
[00:04:07] Anyways, long story short, this is a circle with radius the square root of nine, or in other words, three, centered at zero, zero.
[00:04:31] Let me give you a formula for a circle in general. It is (x minus h) squared plus (y minus k) squared equals r squared.
[00:04:48] So this is a circle with radius r centered at h, k.
[00:05:09] So, write the equation.
[00:05:23] So, this is a circle centered at three, negative two with a radius of five.
[00:05:54] Where does the three go? Five squared, which is twenty-five. That's it.
[00:06:06] Okay, really, it's just a, this is the transformation, the x direction, so to the right by three, up by two, or up by, sorry, up by negative two, which means down by two.
[00:06:21] Let's try another one.
[00:06:34] What is the center and radius?
[00:06:43] So, this is x minus what squared? Zero.
[00:06:53] And so, the center is zero, negative five, and the radius is the square root of seven.
[00:07:11] Okay, not bad. It's always the opposite of this number.
[00:07:22] Can I write down this formula? Yeah. It's x minus the x coordinate, y minus the y coordinate.
[00:07:34] So, it's always just the opposite of that number. So, the opposite is negative five.
[00:07:45] Okay, find the center and radius.
[00:08:10] So, we need to get it in this form. x plus or minus a number, y plus or minus a number, equals some number.
[00:08:24] How are we going to do that?
[00:08:30] First, let's rearrange it.
[00:08:35] So, I just grouped the x's together, the y's together, and brought the constant to the other side.
[00:08:49] I need to take this and factor it to a perfect square binomial. What do we call that process?
[00:09:07] Completing the square. So, we just got to add the correct number to make the same binomial, and so we can write it like that.
[00:09:19] So, how do I complete this square here?
[00:09:26] We take the negative 4 and divide it. Divide by 2 and square.
[00:09:36] And we add that to both sides.
[00:09:38] And then we do the same thing here. Square. 10. Divide by 2. Square.
[00:09:57] That's 25.
[00:10:00] So, the center is the opposite of that number.
[00:10:19] Square root of 26. Just barely larger and higher.
[00:10:25] Okay, let's try another one.
[00:10:49] Okay.
[00:10:51] So, what do we do here? We rearrange everything.
[00:10:56] This is called general form. We want to get into standard form.
[00:11:19] So, we're going to complete the square. So, what do we add?
[00:11:25] 6 divided by 2. 9.
[00:11:38] Negative 12 divided by 2.
[00:12:02] So, we're going to do the same thing here.
[00:12:10] Negative 3, 6.
[00:12:15] And the radius.
[00:12:20] Okay, what questions do we have on this?
[00:12:26] We just have to complete the square twice to figure it out. It's not too complicated.