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CCSS.Math:

we're asked to graph the circle X plus 5 squared plus y minus 5 squared equals 4 I know what you're thinking what's all of the silliness on the right hand side this is actually just the view we use when we're trying to debug things on Khan Academy but we can still do the exercise so it says drag the center point and perimeter of the circle to graph the equation so the first thing we want to think about is well what's the center of this equation well a standard form of a circle is x minus the x-coordinate of the center squared plus y minus the y-coordinate of the center squared is equal to the radius squared so x minus the x-coordinate of the center so the x-coordinate of the center must be negative 5 because the way we can get a positive 5 here's by subtracting a negative 5 so the x-coordinate must be negative 5 and the y-coordinate must be positive 5 because it's y minus the y-coordinate of the center so y coordinate is positive 5 and then the radius squared is going to be equal to 4 so that means that the radius is equal to 2 and the way it's drawn right now I mean we could drag this out like this but this the way it's drawn the radius is indeed equal to 2 and so we're done and I really want to hit the point home of what I just did so let me get my little getting my scratch pad out oh this sorry for knocking the microphone just now so that equation was X plus 5 squared plus y minus 5 squared is equal to 4 squared and so I want to rewrite this as this is X minus negative 5 X minus negative 5 squared plus y minus positive 5 positive 5 squared is equal to and instead of writing it as 4 I'll write it as 2 squared and so this right over here tells us that the center of the circle is going to be NAIT x equals negative 5 y equals 5 and the radius is going to be equal to 2 and once again this is no magic here this is not just you know I don't want to just memorize well you'd appreciate that this formula comes straight out of the Pythagorean theorem straight out of the distance formula which comes out of the Pythagorean theorem remember if you have some Center in this case it's the point negative 5 comma 5 so negative 5 comma 5 and you want to find all of the X's and Y's that are 2 away from it so you want to find all the X's and Y's that are 2 away from it so that would be one of them X comma y this distance is 2 and there's going to be a bunch of them and when you when you plot all of them together you're going to get a circle with radius 2 around that Center let's think about how we got that actual formula well the distance between that coordinate between any of these X's and Y's it could be an x and y here could be an x and y here and this is going to be 2 so we could have our change in X so we have X minus negative 5 so that's our change in X between any point X comma Y and negative 5 comma 5 so our change in our change in x squared plus our change in Y squared so that's going to be Y minus the y-coordinate over here squared is going to is going to be equal to the radius squared so the change in Y it's going to be from this Y to that Y if we this is the end point it'd be the end - the beginning Y - 5 y minus 5 squared and so this shows for any X Y that is 2 away from the center this equation will hold and it becomes I'll just write this in neutral color x + 5 squared plus y minus 5 squared is equal to the radius squared is equal to 2 or let me just write that is equal to 4 and let me make it I I really want to you know I I dislike it when formulas are just memorizing you don't see the connection to other things notice we can construct we can construct a nice little right triangle here so our change in X is that right over there so that is our change in X change in X and our change in Y our change in Y not the change in Y squared but our change in Y is that right over there change in Y our change in Y you could view that as Y minus y so this is change in Y is going to be Y minus 5 and our change in X is X minus negative 5 X minus negative 5 so this is just change in x squared plus change in Y squared is equal to the hypotenuse squared which is the length of which is this radius so once again comes straight out of the Pythagorean theorem hopefully that makes sense