In the following exercises, (a) find the center-radius form of th...

Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 5 and 6.

center (0, 0), radius 6

Accepted Answer

Hello, everyone. We are asked to find the center radius form of the equation of the circle using the given information and plot it on the rectangular coordinate system. We are given that the center is at 00. And the radius is 13, we are given a coordinate plane with an X axis that ranges from negative 15 to positive 15 and a Y axis that ranges from negative 15 to positive 15. There are markers at one unit intervals. First to be able to write the center radius form of the equation, we must know what that is. So the center radius form of the equation is the parentheses, X minus H closed parentheses squared plus the parentheses Y minus K closed parentheses squared equals R squared. So we need to know what these variables stand for. HK stands for the point of our center and R stands for the radius. So here we're told that the center is the 0.00. So this means that H and K are both zeros. We're told that the radius is 13. So let's plug those values into that original equation. So we can get the equation of our circle. So in parentheses, we have X minus H which is zero, closed parentheses squared plus in parentheses, Y minus K which is zero, close parentheses squared equals R which is our radius of 13 squared. I want to simplify this a little bit because X minus zero and Y minus zero don't need to stay like that. X minus zero is just X. So X squared plus Y minus zero is just Y so Y squared equals 13 squared. So our equation is X squared plus Y squared equals 13 squared. Now, we need to try to plot this circle on the rectangular coordinate system. I'm gonna start by plotting the center. The center is the 0.00. It's the origin, it is where the two axes intersect from there. I'm gonna use the radius to make some guide points for my circle. So my radius is 13. So that means the distance from the center to anywhere on the edge of the circle is 13. So I'm gonna start at the center and count up 13 places and put a guide dot And that would be at the 0.0 13, going back to the center. I'm gonna go right 13 places and put a guide dot which would be the 0.13 0 back to the center and go down 13 places and put another guide dot And that would be the 0.0 negative 13. And finally from the center, I'm gonna go left 13 places and I have the point negative 13 0. So now that I have my guide dots in, I'm gonna try to draw a circle that looks like a circle, but bear with me, my circles are never perfect. So we're connecting our guide dots making sure our center stays in the center. And that actually isn't that bad of a circle for me. So we have a circle. A center is at 00. It crosses into all four quadrants and it takes up almost all of the coordinate system we were given if your circle isn't perfect. That's OK. As long as it looks like it fits the points, you have the points in the right spot, no one probably expects you to be able to draw a perfect circle. Have a good day.

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 5 and 6. center (0, 0), radius 6

Key Concepts

Center-Radius Form of a Circle

Graphing a Circle

Distance Formula

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