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

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 (2, 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 told the center is the 0.50. And that we have a radius of 13, we are given a coordinate plane where the X axis ranges from negative 10 to positive 20. And the Y axis ranges from negative 15 to positive 15. We have markings at one unit increments. First to find the center radius form of the equation. We need to know what that form looks like to begin with. So it looks like the parentheses, X minus H squared plus the parentheses Y minus K squared equals R squared. So in this HK is the point for our center and R is our radius. So since we were given that the center is the 0.50 we can say that five is H and zero is K and we're told that the radius is 13. So we have enough information to write this equation. So we have a parenthesis and then X minus H so X minus five closed parentheses squared plus open parentheses. Y minus KK is zero, close parentheses squared equals our radius of 13 squared. I want to simplify this a little bit because Y minus zero is the same as Y. So we're going to rewrite this as the parenthesis, X minus five squared plus Y squared equals 13 squared. So this is our equation. We're done with that part. Now, we need to plot this circle on the coordinate plane. I'm gonna start by plotting the center. So the center is the 0.50. So it's gonna be on our x axis between quadrants one and four. And once I have the center plotted, I'm gonna use the radius. If I can to make some guide points, we might not have enough space because we have a rather large radius. But let's find out. So starting from the center, I'm gonna go up 13 places to find a guide point for my circle and that should be the point 513. So back to the center, I'm gonna go right 13 places to try to put another guide point. And that should be giving me the point 18 0 back to the center. I'm gonna go down 13. So that would be the 0.5 negative 13 and back to the center, we're going to go left 13 places and that's the point negative 80. So now that I have four guide points, I'm gonna use them to try to draw a decent looking circle, but I'm gonna be honest, it doesn't always work out. So let's see how we do. Well, we're a little flat sided, a little lump, lumpy, but we attempted to draw a circle to connect those points and let's see if we can make it a little bit better before we say we're done. All right. That looks a little bit more like a circle. It's hard to draw a perfect circle. But if you have guide points, at least, you know, you're working within the right area. Our circle takes up almost all of the coordinate system we were given and crosses into all four quadrants. 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 (2, 0), radius 6

Key Concepts

Equation of a Circle in Center-Radius Form

Coordinate Geometry and Plotting Points

Radius and Distance in the Plane

Watch next