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^2-2x+12y-12=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. Now, here we are given the equation X squared plus Y squared minus eight X plus 34 Y plus 301 equal to zero. Now, here we have three answer choice options. Answer choice A which states that we have a circle with a center at four negative 17 and a radius of two units. Answer choice B says we have a point with, at the 20.0.4 and negative 17. And lastly, we have answer choice C which states that this circle with such equation doesn't exist. So we want to begin solving this problem by trying to group and completely square of this equation so that we can fit into the standard form of the circle equation. So if we recall the standard form of a circle formula, we have the quantity of X minus H squared plus the quantity of Y minus K squared is equal to R squared. So first we want to start by rearranging our equation so that our X terms are together, our Y terms are together and that our constant is on the right side of our equation. So rearranging, we have X squared minus eight X plus Y squared plus 34 Y is equal to negative 301. So now we want to start completing the square for both our X terms and our Y terms. So beginning with our X terms, we have the quantity of X squared minus eight X. And now we need to take the coefficient of our X value and divided by two and then square the quantity to completely square. So we have plus the quantity of eight divided by two squared. And next, we have plus our Y terms, we have plus the quantity of Y squared plus Y. And now repeating the completing the square process, we also have plus 34 divided by two quantity squared. And this full expression is equal to negative 301. And now, since we added these values on the left side of our equation, we also need to add them to the right side of the equation. So we have negative 301 plus the quantity of eight divided By two squared plus the quantity of divided by two squared. So now we just want to start simplifying our expression on both sides by dividing each of the appropriate quantities by two. So we have X squared minus eight X plus four squared. And we're going to group this expression in parentheses. And then we have plus Y squared plus Y plus 17 squared is equal to negative 301 plus four squared plus 17 squared. And now we want to continue to simplify by squaring each of the appropriate terms. So we have the quantity of X squared minus eight X plus 16 plus the quantity of Y squared plus 34 Y plus 289. And this is equal to negative 301 plus 16 plus 289. And now we have finished completing the square so we can rewrite our group terms as a squared expression. And then we just want to simplify the some on the right side of our equation. So now our equation becomes the quantity of X minus four squared plus the quantity of Y plus 17 squared is equal to four. And now we want to rewrite the term on the right side of our equation, which is for as a squared quantity so that it matches our formula where we have R squared. So doing this, our final equation becomes the quantity of X minus four squared plus the quantity of why plus squared is equal to two squared. So now looking at the equation that we came up with here, we see that we do in fact have a circle because our equation now matches our circle formula. So we know that the of a circle is the point H K. Well, the values for H K in our equation are and negative 17. And now we also know from our formula that the right side is R squared. So our radius is the value for R. So the R value or radius is two units. So just to summarize our equation tells us that we do have a circle and we see that we have a center at -17 and our radius is two units. So with this, what we found we are left with answer choice. A where we have a circle with a center at 474 and negative 17 and a radius of two units. 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^2-2x+12y-12=0

Key Concepts

Standard Form of a Circle

Completing the Square

Types of Graphs: Circle, Point, or Nonexistent

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