SM2 14.1-4: Given the Circle Center and a Point, Write the Equation

Added: 2026-05-10

[00:00:00]Hello and welcome. So we are given the center of  a certain circle is 6 comma 10 and the point the circle, a point on the circle that uh the circle  passes through is negative 9, 9. What is the equation of the circle? So essentially what  happens is that we're given some, I'm not going to put a coordinate plane, we're given some  circle, okay? And we're told that the center is going to be at 6 comma 10. So whatever this  point is at, is 6 comma 10. That's its location.

[00:00:29]Then we're told that we have a point at like  negative 9 comma 9. That's like maybe right around there, right? This is a point on the circle. When  it says when it passes through it just means that there's a point on the circle. It's at negative  9 comma 9. That's that's as much as you need to know. Okay. Now in order to write the equation for  this thing our general equation for any circle is x minus h squared plus y minus k squared equals your radius  squared. All right, so when we're looking at this thing we've got to know three things: our h,  our k, and the r, the radius. Well the h and k is not that bad because any center of a circle is  literally the h comma k. So if you know the center, you know h and k. Do you know the center? Yes we  do. What a coincidence, right? There you have it.

[00:01:17]So that means that our center is 6 comma 10. That  means that your h is 6 and your 10 is k. So right then and there we have six-- whoops, apologies, x minus 6 squared plus y minus 10 squared equals our radius squared. Bam.

[00:01:33]Okay, and now we just got to find the point the  circle passes through. So in other words how do we use the second piece of information to find  our last bit of information we need, the radius?

[00:01:44]Well think about it this way. We know that  negative 9, 9 is a coordinate point and that is x comma y. To find that point, well that  means that your x has to be negative 9 when your y is 9. We have three variables inside of our  equation. We have x, y, and r. And if we know that, that these two points, we have negative 9 and 9,  all right? That is your x and y. And that would leave only one variable left, the r, your radius.  So then we have our x, negative 9, plus our y, 9, equals our radius squared. And you'll notice  that bam, this is most of the legwork because there's only one variable left. It's your r. All  you have to do is simplify the numbers that you have and you're good to go. So this is now order  of operations where it's negative 9 minus 6 which is negative 15, plus 9 minus 10, negative  1, equals your r squared. And now we just do those squared. So then 15 squared is 225. Now the reason  why I say it 15 squared is because when you have a negative 15 squared that's negative 15 times  negative 15. That's positive 225, right? And then plus negative 1 squared, 1, equals r squared.  225 plus 1 is 226. Now do we have to solve for r?

[00:03:11]No. Think about it this way. The only thing  you're looking for is that you need to know r squared because that's what needs to go in your  equation. You have solved for r squared. r squared equals 226. You don't have to do anything further  than that, and you're essentially done with your equation because all we're doing now is we're  plugging that into our initial equation there and we're done. So we have x minus 6 squared  plus y minus 10 squared equals 226. Right, the number we just found. That is our r squared.  And that is the equation of the circle that has a center at 6 comma 10 that passes through the  point negative 9, 9. Thanks for watching.