Use each graph to determine an equation of the circle in (a) c...

Question

Use each graph to determine an equation of the circle in (a) center-radius form and (b) general form.

Graph of a circle centered at (1,1) passing through points (3,3) and (5,1) on an xy-coordinate plane.

Accepted Answer

For the given circle, write the equation that best describes it and write or answer in standard form and general form. And we know this here, we have a circle with four points. So we need to write this in standard form and general form. Let's first find the center of our circle to find the center. We need to point H K and we can find H by adding our X end points here and finding the midpoints between this by dividing it by two. So in other words, let's say H midpoint, we'll use our one comma three Y comma three points add those together divided by two. That is 10 divided by two, which is five or K will do the same thing. But with R Y and the points at the circle, we notice we have five negative one, N 57. OK. Then will be negative one plus seven divided by two. Give me a six divided by two, which is three, our center then is the 30.53. We can also by the radius, the radius will be the distance from the center of a circle to any of the end points of the circle. For instance, we had 53 to get to the 0.93, we will go on the X four, our radius then should be four. Now let's write this in standard form. Standard form of a circle is given by the form X minus H squared plus Y minus K squared equals R squared. We have H K and R so we can plug them into our equation X minus five squared plus Y minus three squared equals four square could simplify then take X minus five squared plus Y minus three squared equals 16. Now, we have our standard form, we can write this in general form to write in general form. He wants to move everything to one side and set equal to zero as well as expand our two exponents. Here, you have X minus five squared. This expands to X squared minus 10, X plus 25 Y minus three squared, expands to Y squared minus six Y plus nine set. This equals to 16. We will then move our 16 over to the left side getting X squared minus 10, X plus 25 plus Y squared minus six, Y plus nine minus 16 equals zero. The now combine our 25 9 and 16, bring down our X squared and X are Y squared and minus six Y, we can combine a 25 9 in to get plus 18. We can't simplify anymore. So we have our general form here, our standard form X minus five squared plus Y minus three squared equals 16. And our general form X squared minus 10, X plus Y squared minus six Y plus 18. If you look at our possible answers, we can determine answer. D has both forms that we found. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

Use each graph to determine an equation of the circle in (a) center-radius form and (b) general form.

Key Concepts

Center-Radius Form of a Circle

Finding the Center and Radius from a Graph

General Form of a Circle's Equation

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