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# Features of a circle from its expanded equation

CCSS.Math:

## Video transcript

we're asked to graph the circle and they give us this the somewhat crazy looking equation and then we could graph it right over here and to graph a circle you have to know where its center is and you have to know and you have to know what its radius is so let me see if I can change and you have to know it's radius is so what we need to do is put this in some form where we can pick out its center and its radius so let me get my little scratch pad out and see if we can do that so this is that same equation and what I essentially want to do is I want to complete the square in terms of X and complete the square in terms of Y to put it into a form that we can recognize so first let's take all of the X terms so let's take all of the X terms so you have x squared + 4 X on the left hand side so I could rewrite this as x squared + 4 X and I'm gonna put some parentheses around here because I'm going to complete the square and then I have my Y terms I'll circle those in well the red looks too much like the purple I'll Circle those in blue y squared and negative 4y so we have plus plus y squared minus 4y and then we have a minus 17 and I'll just do that in a neutral color so minus 17 is equal to 0 now what I want to do is make each of these purple expressions perfect squares so how could I do that here well this would be a perfect square if I took half of this 4 and I squared it so if I made this plus 4 then this entire expression would be X plus 2 squared and you can verify that if you like we've if if you need to review on completing the square there's plenty of videos on Khan Academy on that all we did is we took half of this coefficient we took half of this coefficient and then squared it to get 4 half of 4 is 2 squared to get 4 and that comes straight out of the idea if you take x + 2 and square it it's going to be x squared plus twice the product of 2 and X plus 2 squared now we can't just really nearly add a 4 here we had an equality before and just adding a 4 wouldn't it wouldn't be equal anymore so if we want to maintain the equality we have to add 4 on the right hand side well now let's do the same thing for the Y's half of this coefficient right over here is a negative two if we square a negative two it becomes a positive four we can't just do that on the left hand side we have to do that on the right hand side as well now we have in blue becomes Y minus two squared and of course we have the minus 17 but why don't we add 17 to both sides as well to get rid of this minus 17 here so let's add 17 on the left and add 17 on the right so on the left we're just left with these two expressions and on the right we have 4 plus 4 plus 17 well that's 8 plus 17 which is equal to 25 now this is a form that we recognize if you have the form X minus a squared plus y minus B squared is equal to R squared we know that the center we know that the center is at the point a comma B essentially the point that makes both of these equal to 0 and that the radius the radius is going to be R so if we look over here what is our a we have to be careful here our a isn't to our a is negative 2x minus negative 2 is equal to 2 so the x coordinate of our Center is going to be negative 2 and the y coordinate of our Center is going to be 2 remember we care about the x value that makes this 0 and the y value that makes this 0 so the center is negative 2 comma 2 and this is the radius squared so the radius is equal to 5 radius is equal to 5 so let's go back to the exercise and actually plot this so it's negative 2 comma 2 so our Center is negative 2 comma 2 so that's right over there X is negative 2 y is positive 2 and the radius is 5 so see this would be 1 2 3 4 5 shifted a little bit wider than this my my little pen is having trouble there you go 1 2 3 4 5 let's check our answer we got it right