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Determine the center and radius of the given circle and sketch its graph.$$2 x^{2}+2 y^{2}-6 x-10 y+9=0$$

$$\mathrm{C}(3 / 2,5 / 2) \mathrm{r}=2$$

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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 the ra…

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Find the center and radius…

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for this problem. We're examining the given equation two X squared plus two y squared minus six x minus 10 y plus nine equals zero. This is the equation of a circle. So in order to graph this, we need to find both the center and the radius. Now, if we were given this in standard form, that would be easy. Let's do a quick review. Standard form of a circle is X minus h squared. Plus why minus k squared equals R squared. Standard form is fantastic for sketching circles because everything you need is visual visibly available to you. The constant term is our square, so it's take the square root, you have your radius, and the center of the circle is going to be at the point HK, so everything you need is right there. Unfortunately, that is not what we were given. What we're given is that the general form where everything is set equal to zero now we can go from one to the other. So let's go from general form to the standard form and then we'll be able to graph it. First of all, if you notice in our standard form, it's X and why there's no coefficient in front of them. So I'm going to divide everything here by two and get these two just x and y so it becomes negative. Three X minus five y plus nine haps. Okay, now that we have just x squared and why squared, we're gonna need to complete this square. So I'm going to collect all of my ex terms together, all of my y terms together and that nine halves, that constant, I'm going to push to the other side of the equal sign. So be a negative nine halves. Yeah, so let's complete the square. First we look at the X term. Okay, so we have the negative three. We take half of that, which is negative. Three halves and we square. It's we're gonna add 9/4 and we need to do that to both sides to make sure our equation stays balanced. So that becomes X minus three halves squared. Now for our wise again, we look at the Y term. We take half of that coefficient, negative five halves, and we square it plus 25 4th and we add that to both sides. So that gives us completing the square for the why gives us why minus five halves squared. Now, what about our constant? Well, we need to come and denominator. So this negative nine halves, I can write his negative 18 4th. And when I add those altogether, I end up with 16 4th, which is four. Okay, so what does this tell me? Well, constant is four. Take the square root of a radius of two. And what? Um, I subtracting from X and why that's my center. Three halves, five halves. So if I take three halves, five halves, that puts it right about there. I've got a little blue dot and I want to go to two units in either direction. Connect those dots there, and that is the sketch of a circle with a center at three halves, five halves and a radius of two.

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