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Determine the center and radius of the given circle and sketch its graph.$$9 x^{2}+9 y^{2}-12 x+24 y-101=0$$

$$\mathrm{C}(2 / 3,-4 / 3) \mathrm{r}=11 / 3$$

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

Missouri State University

McMaster University

Harvey Mudd College

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Determine the center and r…

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Find the center and radius…

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for this problem, we're going to examine the equation. Nine X squared plus nine y squared minus 12 X plus 24 Y minus 101 equals zero. This is the equation of a circle. So in order to graph the circle, what we need is the center and the radius of the circle. Now, if we had this in standard form, this would be very, very simple. Remember, standard form looks like this X minus h squared. Plus why minus k squared equals R squared. So this constant term is the radius squared so I could take the square root, find the radius of my circle, and my center is at the point HK The numbers I'm subtracting from that X and Y, respectively. So if I have this in standard form, I can visually just see the radius. I can see the center and I could grab it. Unfortunately, what we have been given is not standard form. It's general form where everything is set equal to zero. So our goal for this problem is to go from general form to standard form so that we can graph our circle. Well, first of all, if you look at the standard form, it is just X minus. And why minus no coefficients. So I want to get rid of that nine in front of my X squared and Y squared terms. So I'm going to divide everything by nine. That gives me X squared. Plus Y squared 12/9 is four thirds x 24/9 is eight thirds. Why minus. And this is not, um, factor at all. Minus 100 won over nine. Now, we're gonna complete the square once for the exes. So I'm gonna pull my ex terms together and once for the why. So I'm going to pull my Y terms together, my constant. I'm going to move to the other side of the equation. Now let's complete the square. First, our exes, we take the X term, we take half of its coefficient, which is negative two thirds, and we square that four nights. We can add that to both sides of our equation. So what that gives me is X minus two thirds squared. Now for the wise. I look at my y term, half the coefficient. It's four thirds square that I get 16 3rd and I can add that as long as I do it to both sides, that gives me why. Plus four thirds squared. And if I add everything over here, I do need I'm sorry. I don't know why I just wrote three here at 16. Over nine. I'm I apologize for that. 16/9 is what I get when I square four thirds. So I'm adding 16/9 to both sides. I have a common denominator, and that gives me 121 over nine. Okay, Not the prettiest numbers ever. But we can make this work our constant is our radius squared. If I take the square root of that, I get 11 3rd. So when I go to graph that, that's just under 4. 12 3rd would have been four. The center. When I'm subtracting from X and y, I'm subtracting two thirds from X and I'm subtracting a negative four thirds from y. Remember, if it's a plus, why plus something? It's become a negative when I take the coordinates for the center. Okay. Now let's plot those points. If I take the 0.2 3rd negative four thirds, it is approximately right there, and I'm going out approximately 4 11 3rd in each direction. 1234 So right about there 1234 Right about there. 123 Right about there. 123 Right about there again. It's a sketch, but that is an approximate sketch of the circle that has a center at two thirds negative four thirds and a radius of 11 3rd.

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