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The Circle - Problem 3
Start by writing the given equation of a circle in standard form: r^2 = (x-h)^2 + (y-k)^2. This may require completing the square. Gather the x variables and the y variables on one side of the equal sign, and the constant to the other side. Next, complete the square for both variables. Whatever is being added to one side when completing the square should also be added to the other side to keep the equation balanced. Once the equation is in standard form, you can determine the coordinates of the center and the radius by look at the h, k, and r values. To find the coordinate of the center, extract the h and k values. Since the quantities are (x-h) and (y-k), that means that the value of just "h" and "k" will have the opposite sign. In other words, if the quantity is (x-1), then h = 1. If the quantity is (x+3), then h = -3. Since r is being squared, the radius is the square root of the constant value. In other words, if r^2= 25, the radius is 5.
So we're good at finding the center and the radius for a circle when we have it in x² plus y² is equal to r² form, but we're not always going to get equations in that form. So like what I have behind me is just this string of information. What we want to do is get it into the form where we can recognize what we're looking at and the way to do that is by completing the square, our favorite.
So what we want to do is isolate our x's together, isolate our y's together and get the constant term to the other side by itself.
So just doing a little bit of rearranging, what I'm going to do is x² minus 2x leave a little space to complete the square there, plus y² plus 6y leave a little space after that and now this is equal to 15. So now we just want to complete the square, x remember we divide our middle term by 2 so -2 divided by 2 is -1 that is what goes here, -1² that is what ends up going over here. So we added one into this parenthesis, so to keep it balanced, we have to add one over to the other side as well.
Doing the same thing with the y's plus again divide our middle term by 2, so 6 divided by 2 is 3 and that ends up being squared, so we actually end up adding 9 keeping the sides balanced again we have to add 9 to the other side, and so what we end up with on the right is 15 plus 1, 16 plus 9, 25.
By completing the square, we were able to get our equation just from a string of information into a form that we recognize, so now all we have to do is extract the information we need. We're asked for the center, so x² minus the center is 1, y minus the center that will be -3 so our center is 1,3 and that's equal to our radius 25 square root of that is 5 so the radius is equal to 5.
So finding the center and radius of a circle by completing the square.