 In this video, write an equation of a circle. We are given three problems and we're asked to write an equation for the circle with the given information. These problems are similar to the practice problems one that we just did where we were shown a graph and asked to write an equation of the circle and now this time we're just given the information and asked to plug it into our standard equation. So the first one is fairly straightforward. We're given our center at x equals 3, y equals negative 1, and a radius of 4. We don't need to graph these or do anything else, we just need to plug that information into our standard equation of a circle, remembering that we're going to change the sign of the x and y values of the center of our circle so that positive 3 becomes negative 3 squared and the negative 1 up here becomes positive 1 squared. And then our radius of 4, we're going to square that. And so to clean this up, all we need to do is leave the left hand side the same. That's the form that we want to keep it in for the equation of the circle and then just simplify 4 squared which is 16. And there's the answer to part A and we'll move on to the second problem. Similar situation here with just a couple differences. The radius you'll notice is a mixed radical that we'll deal with and it says that our circle is centered at the origin so I'm just going to write up here my center point if it's centered at the origin is just 0, 0. So when I plug that information in these 0's, I can write them if I want but when I clean it up you'll see that we'll just make those go away. So technically I can put those in. My radius 3 root 2 squared, I'm going to remember that when I square a mixed radical, I'm just going to do this off to the side here. If I do 3 root 2 times 3 root 2, I'm going to multiply the 3's, 3 times 3 is 9. And then root 2 times root 2, the radicals cancel out and I'm left with just 2. Anytime you square a square root, the radical symbol cancels out so 9 times 2 is 18. The right side of this equation, my radius squared is going to be 18. And then I'm just going to clean up the other side because when we have a circle that is centered at the origin, we can just write it like this. And that's the final answer for the second problem. When we come to the third problem, this one is a little different because we are given the center point at the origin so we know what to do with that. But we're not given a radius. We're just told that this circle goes through this point. And so we have to do some work to find the radius of this circle. And there's a couple different ways you can do that. I'm going to show you both ways and then you can decide which one works better for you. You don't need to graph this information. I'm going to just show you the graph of this though so we can get a visual on what they're asking us to do. Again, the circle is centered at the origin so I put my center point at the origin. And then the second piece of information, the circle goes through point one two. I just graphed point one two on the coordinate graph and then drew the circle around that so I could get a visual of what they're asking me. Remember I need to find the radius and so I need to find this distance from the center point out to that given point on the circle. And then that kind of gives you a clue of what one method is we could use. If I say I have to find this distance to find the radius of course we can use the distance formula. So for the distance formula I have two points. I have this point on the circle and I have my origin which the x, y values of that point is zero zero. Anytime I have two points I can find the distance by finding the difference of the x's one minus zero squared and then adding the difference of the y's squared. Remember we take the square root of that and when we simplify that we're going to get one squared which is one plus two squared and one plus four we have a distance of the square root of five. That represents this distance from the center point up to that point on my circle and we know that anytime we have a segment that's from the center point to the edge of the circle that's going to be my radius. So now I have a radius of root five but before I finish this problem I'm just going to show you one other way if you don't like using the distance formula to find that answer of root five. If I bring up another graph again remember I'm trying to find this distance right here and anytime we have two points on an actual graph I can make a right triangle and we all know if I can make that line we all now know how to solve missing pieces of a right triangle if we're given two sides we can find the third side with a Pythagorean theorem and we are given this side of one because that's one unit long for the short leg and then if we count up the long leg is going to be two units long and so in order to find this radius we're just going to set that up as a hypotenuse and we're going to do the Pythagorean theorem side squared plus side squared equals the hypotenuse squared and when we simplify that you'll see we get a similar result because the distance formula is going to give us the same results as the Pythagorean theorem. So now that I have that radius I can go back to my problem and fill in my information I know that my radius is root five and the center point of my circle is zero zero so now we're just going to plug that in like we did before remember you don't need to put the zero point in I'm just doing that before I simplify and remember we're going to square that radius and so when I clean that up my final answer for the equation of this circle with the given information x squared plus y squared when I have the square root of five squared when I simplify that root five times root five is just five remember this does not mean that the radius of five is five it just means when I put that in the standard equation of a circle we need to square that and that is my final answer.