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

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

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

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University of Michigan - Ann Arbor

Idaho State University

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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for this problem. We're going to examine that given equation that I have written down here. Now, this is the equation of a circle. So in order to sketch it, we're going to need to find both its center and its radius. Now, to do that, we're gonna want this in standard form. Standard form looks like this. And I'll do it down here at the bottom to myself. Lots of room for the actual doing. The problem Standard form says I have X minus h squared. Plus why minus k squared equals R squared. The constant is our radius squared will take the square root to get the radius And HK is going to be my center Whatever I'm subtracting from X and y now that's our standard for what we've been given is general form where it's set equal to zero. They both are have their uses, But the standard for miss so nice, because by inspection you confined both the center and the radius. So let's take our general form and put it into standard form. Now, in order to do that, we need to complete the square. So I'm going to collect all of my exes together and all of my wise together, I haven't changed. Just rearranging the order a bit. Anything that is neither x nor why I'm gonna move over to the right hand side. I'm just going to subtract nine. I'm gonna leave the constant term over on the right hand side. Eventually, that's going to show me my radius. Okay, Now we're going to complete the square. If you remember how this works, you take half of that X term, which in this case is three. And you square it. That's what you're gonna adhere to complete the square. I'm missing a plus nine, and I'm going to do it to both sides of the equation. That way I stay balanced. So what this really gives me now for this first piece is X plus three squared. Well, what about my next one? What about the wise? Well, again, I take half of my the coefficient of my Y term, which is negative. Five. I square it. That gives me 25. And again I have to add that to both sides in order to stay balanced. So what That looks like when I actually put that together into a perfect square. It's why minus five squared, and that's gonna equal while my ninth we're going to cancel So it will be left with is 25. There is my standard form. So what does this tell me? I'll do this and blues what stands out a little bit? 25. That's my radius squared so my radius will be five. And if I look at what I'm subtracting for my center X Plus three I can think of that is X minus negative three. And why minus five remembered the signs of within the equation or opposite the signs of the coordinates. X Plus three is a minus three as a coordinate. Why minus five is a positive five for that center. Coordinate. So now let's get your circle. I have the point. Negative. 312345 There's my center with a radius of five. I'm going to go out five units in all four Cardinal directions. 1234 yet and I'm just going to connect those dots. And it's not gonna be a perfect circle because I'm on a white board here. But it's a fairly good sketch to show that we do have a circle. It has a center at negative 35 and a radius of five

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