Connecting Graphs with Equations Use each graph to determine an e...

Question

Connecting Graphs with Equations Use each graph to determine an equation of the circle in center-radius form.

Accepted Answer

Hello, everyone. We are asked to write the center radius form of the equation of the circle using the graph. We are given a circle on a coordinate plane. It is mostly in quadrant four but does cross the positive X axis into quadrant one. There are four points listed on our circle. The 0.17 1 23 negative five, 17, negative 11 and 11 negative five. Our answer choices are a, the parentheses, X plus 17 squared plus the parentheses, Y minus five squared equals six squared. B, the parentheses X plus 17 squared plus the parentheses, Y plus five squared equals six squared C, the parentheses X minus 17 squared plus the parentheses, Y plus five squared equals six squared D. The parentheses X minus 17 squared plus the parentheses, Y minus five squared equals six squared. The first thing to recall to be able to write this in center radius form is what center radius form is. So center radius form is the parentheses, X minus H closed parentheses squared plus the parentheses, Y minus K closed parentheses squared equals R squared where HK is the center of our circle and R is our radius. So right now, we're given points, but we have to figure out what is the center of this circle? We aren't directly told. However, we can use any points, set of points that are directly across from each other and use that to find the center. So I'm gonna use the points 17, 1 and 17, negative 11. Since those points are directly on top of each other and on the same line, I can add the X's and divide in half to find the X value of the center. So I'm gonna do that. That's gonna be H so H is the X values added together. So 17 plus 17 and then divided by two. So 34 divided by two, which is 17. So we know the H part of our center for K, we're gonna do the same thing with the Y values, gonna add the Y values together. So one plus negative 11 and then we'll divide it by two. So let's see, one plus negative 11 is negative 10, divided by two is negative five. So we now know the center of our circle is the 0.17 negative five to find the radius. We're gonna use the distance formula. So we're going to use the distance formula, which recall is that distance equals the square root of X one minus X two squared plus in parentheses, Y one minus Y two again squared. So since I only want the radius, I'm gonna find the distance from one of these points that I used already to the center. So we're gonna use it from the point 17 1 to the center which we just found was 17 negative five. So I'll call 17 1 X one Y one and 17 negative five X two Y two. So our distance here is the square root of in parentheses 17 minus 17 squared. Thus the parentheses one minus negative five squared. Simplifying that 17 minus 17 is 00 squared is zero. So I have the square root of zero L one minus negative five, double negatives are like adding. So that's like six squared which is 36. So I have the square to 36 which is six. So this means our radius where the distance from a point to the center is six. So now that I know HK and R, I'm gonna go back to our center radius formula and in parentheses I have X minus H so H is the X value from the center. So 17 closed parentheses squared plus open parentheses, Y minus K which is the Y value of the center. So Y minus negative five close parenthesis squared as to equal our radius, which we found was six squared. So the only thing to clean up here is that I have a double negative in my second set of parentheses. So I'm gonna rewrite this as the parentheses, X minus 17 closed parentheses squared plus the parentheses Y plus five closed parentheses squared equals six squared. So this is our equation and that appears to match answer choice. C have a good day.

Connecting Graphs with Equations Use each graph to determine an equation of the circle in center-radius form.

Key Concepts

Circle Equation in Center-Radius Form

Graphical Interpretation of Circles

Coordinate Geometry

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