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, 5), 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 plotted on the rectangular coordinate system. We are told the center is the point negative 38 and the radius is six, we are given a coordinate plane where each interval is one and the positive X axis goes to 15. The positive Y axis goes to 15, the negative X axis goes to negative 15 and the negative Y axis goes to negative 15. First, I'm going to find the center radius form of the equation. Recall that the form for that is that we have a set of parentheses, X minus H closed parentheses squared plus the parentheses Y minus K close parenthesis squared equals R squared where HK is the point for the center and R is the radius. So it looks like we were given all of this information. So our center we're told is negative 38. So that means negative three is H and eight is K and we're told that our radius is six. So I'm gonna plug those values into the equation formula or format. So I have a parenthesis then I have X minus H is negative three. So X minus negative three and I put that negative three in parentheses squared plus the parentheses, Y minus KK is eight. So Y minus eight in parentheses squared equals the radius which is six squared. I don't want to leave a double negative in my first set of parentheses. So I want to rewrite that as the parentheses, X plus three, close parenthesis squared plus the parentheses Y minus eight, close parenthesis squared equals six squared. So we have our equation. Now we wanna plot this circle on the coordinate plane. The first thing I'm gonna plot is the center. So negative 38. So negative three on the X and up eight. So our center lands us in the second quadrant from there. Since I know the radius, I'm gonna go up six blocks from the center and put a dot Just as a guide. So one 456 that lands me at the point negative 314. Then I'm going back to my center and I'm gonna go down six and that lands me at the point negative 32, going back to the center, I'm gonna go right six, put a point and that's at the 0.3 um eight. And then going back to the center, I'm gonna go left six and that lands me at the point negative 98. So now that I've put guide points, I'm gonna try my best to make a circle that looks like a circle. Well, it's not perfect but it looks enough like a circle. Um We have the center at negative 38. The circle is predominantly in the second quadrant but is a little bit in the first quadrant. It crosses the X axis or sorry, the y axis but does not cross the X axis. Maybe your circle is rounder than mine. But we can always try again. Have a nice 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, 5), radius 4

Key Concepts

Equation of a Circle in Center-Radius Form

Understanding Coordinates of the Center

Graphing a Circle on the Coordinate Plane

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