Describe the graph of each equation as a circle, a point, or none...

Question

Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. See Examples 3–5. x^2+y^2+4x+4y+8=0

Accepted Answer

Hey, everyone. Here we are asked to identify whether the following equation describes a circle or a point. Or if it doesn't exist here, we are given the equation X squared plus Y squared plus 18, X plus 18, Y plus 162 is equal to zero. Now, here we have three answer choice options. Answer choice A which states that we do in fact have a circle with a center at negative nine negative nine and a radius of nine units. Answer choice B says we have a point which is the point negative nine and negative nine. And answer choice C states that the circle does not exist. So our first step is to recall the standard form of the circle equation which is the quantity of X minus H squared plus the quantity of why minus K squared is equal to R squared. Now, we just want to rearrange our given equation and try completing the square so that we can try to fit this equation as the standard form of the circle formula. So rearranging, we want to group together the X terms and the Y terms and then have our constant on the right side of our equation. So rearranging, we have X squared plus 18 X plus Y squared plus 18 Y is equal to negative 162. So now that we have grouped and rearranged our equation, we want to start completing the square for both the X variables and the Y variables. So starting with the X terms, we have the quantity of X squared plus 18 X and now we have plus, and we take the coefficient of X which is 18, then we divide it by two and take the square of that quantity. And now we repeat this with our white terms. So we have plus the quantity of Y squared plus 18 Y plus the quantity of divided by two squared equal to negative 162. Now, since we added values to the left side of our equation, we have to repeat this on the right side of our equation. So we have negative 162 plus the quantity of 18 divided by two squared plus the quantity of 18 divided by two squared. So now we just want to start simplifying our expression. So we have the quantity of X squared plus 18 X plus 9, 18 divided by two, gives us nine. So we have nine squared plus the quantity of Y squared plus 18 Y plus nine squared. And this is equal to negative plus nine squared. Plus another nine squared. Now, we just want to square all of the appropriate terms. So we have the quantity of X squared plus X plus 81 plus the quantity of Y squared plus 18 Y plus 81. And this is equal to negative 162 plus 81 plus 81. So now we have finished completing the square. So on the left side of our equation, we can now rewrite our group, the X terms and Y terms as a squared expression. And then on the right side, we just want to continue to simplify our song. So we have the quantity of X plus nine squared plus the quantity of Y plus nine squared is equal to negative 162 plus 162. Now here simplifying further, we see that the right side of our equation cancels out. So we have the quantity of X plus nine squared plus the quantity of Y plus nine squared is equal to zero. Now, since our right side of the equation is zero, this tells us from our formula that the radius is zero. So if the radius is zero, we cannot have a circle. But instead we have a point. Well, the coordinates of the point are where X and Y are both equal to zero. So when X and Y are both equal to zero, the X and Y values or not X and Y being equal to zero. But the expression Y plus nine and X plus nine must both simultaneously be equal to zero. So this tells us that the values for X and Y that make the right hand side of our equation zero are the coordinates of the point which will be at negative nine and negative nine. So therefore, we are left here with answer choice B where our equation represents a point at the coordinate points negative nine, negative nine. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. See Examples 3–5. x^2+y^2+4x+4y+8=0

Key Concepts

Standard Form of a Circle

Completing the Square

Types of Graphs: Circle, Point, or Nonexistent

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