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 (0, 4), radius 4

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 that the center of the circle is the 0.0 11. And then it has a radius of seven. We were given a coordinate plane where the X axis ranges from positive 15 to negative 15 and the Y axis ranges from positive 20 to negative 10. They are marked in one unit increments. So first, we want to find the center radius form of the equation. So first thing we should find is what is the formula for that? What is our normal format? So the normal format for that is we have the parentheses X minus H squared plus the parenthesis Y minus K squared equals the radius squared. So in this, we need to know that HK is the center of our circle and R is our radius. So we were given that our center here is the 0.0 11. So this means that zero is our H 11 is our K and we were given the radius is seven. So let's plug those things into this formula and then simplify if needed. So we have the parentheses, H, I'm sorry, the parentheses X minus H so X minus zero, close parentheses squared plus the parentheses, Y minus K. So Y minus 11, close parentheses squared equals our radius, which is seven squared. So the only thing here I can see that needs to be simplified is that X minus zero is the same as X. So we're gonna rewrite this as X squared plus the parentheses Y minus 11 squared equals seven squared. So this is our equation. We don't need to do anything else with that. Right now, we want to plot the circle on the coordinate system. Now, I know I cannot draw a perfect circle so bear with me. But the first thing I'm gonna plot is the center. So 0 11, so zero on the X 11 on the Y axis. So right now our center is between is on the Y axis, between quadrant one and quadrant two. Since I know the radius, I'm gonna use this to set a gauge for myself. I'm gonna put some guide points. So from the center, my radius is seven. So I can go up seven spots and put a guide dot So that lands me at the 0.0 18, going back to the center, I'm going to go right seven places to put another guide spot and that lands me at the point 7-Eleven back to the center. I'm going to go down seven that lands me at the 0.04 and back to the center. I'm gonna go left seven for one more guide location. And that would be the point negative 7-Eleven. So now that I know where the center is, and I've kind of given myself a guide to make this circle, I'm gonna try to make a round circle. Well, connecting the dots. I got a little lopsided but our circle is half in quadrant, one half in quadrant two. It does not touch the X axis and the Y axis splits it directly in half vertically. Maybe your circle's rounder than mine either way. 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, 4), radius 4

Key Concepts

Equation of a Circle in Center-Radius Form

Understanding Coordinates of the Center

Graphing Circles on the Coordinate Plane

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