Okay, the equation of a circle with endpoints of a diameter at 2, 3, and negative 4, 5. Okay, equation of the circle. Again, we're after x minus h squared plus y minus k squared equals r squared. Now, the end points of the diameter, so let's just kind of get a ballpark of what's going on here.
Just a rough sketch. So the diameter end points are at 2, 3, and negative 4, negative 5. Somewhere over there. So, just trying to get a rough picture. It's not going to be perfect there, but I'm just looking for a rough. Not even necessarily going through the origin in that look.
But I just want to see this, this is the diameter. So you can tell that the circle is roughly... You know of that, the center is somewhere right in here. So there's a couple of concepts we can work on.
We need to know where the center is, that's the H and the K, and we also need to know how long the radius is. So there's a couple of concepts at play here. Let's just take one at a time. Let's work on the center.
Okay. The center here is actually the midpoint. of the diameter, the midpoint of the diameter.
So midpoint, recall the formula, add up the x's, divide by 2, add up the y's, and divide by 2. So since I know these coordinates, this one was negative 4, negative 5, this one was 2, 3, well I can add up the x's, so let's put that together. There's a negative 4 and a 2, and then there's a negative for the y's, there's a negative 5 and a 3. So in this first one we're looking at the x coordinate is negative 2 over 2 and this one is negative 2 over 2. So we're looking at a center of negative 1, negative 1. Let me write that out there again, center. And keep in mind this is the h and the k values.
Now the next thing we need to do is find the radius. So the radius, let me just take a note here. The radius would be, you could say, there's a couple of things you can do.
You could say it's half the length of the diameter. Or you could say it's the distance from the center to one of these. You know, and it really doesn't matter. I think probably the easiest thing to do would be maybe if we know that this center is at negative 1, negative 1, we'll just find that distance across there, and that's our radius. So we've got a point negative 1, negative 1, and a point 2, 3. And we use the distance formula.
Okay, so keep in mind the distance formula here is x2 minus x1 squared and y2 minus y1 squared. Take the square root of that. So substituting in these given values. We've got an x two, I'll just call that two. Two minus, and then the x one is negative one.
Don't forget to square that. y2 is 3 minus another negative 1. Square that. So we're looking at 2 plus 1, so 3. We're going to square it. 3 plus 1 is 4. We're going to square it.
That becomes nine and sixteen, which makes twenty-five. So you want the square root of twenty-five is five. And that's the distance from negative one, negative one to two, three, which is the radius. I'll just write is the radius. So, r squared is 5 squared, which is 25. So back to this formula that we had up here for the equation of the circle.
It says, what is the equation? We found the center. We found the radius.
Now we substitute them in. And my final answer is x minus a negative 1. squared plus y minus a negative 1 squared equals r squared. You can say 5 squared or 25. So simplified x plus 1 squared plus y plus 1 squared equals 25. There's the equation of the circle. Center at negative 1, negative 1. Practice that. Radius of 5.