Work each problem. Write the inequality that represents the regio...

Question

Work each problem. Write the inequality that represents the region inside a circle with center (-5, -2) 
and radius 4.

Accepted Answer

Hello, today we're gonna be finding the equation of the circle with the given information. So we are told that the center of a circle is located at negative three and negative four. And we are told that the radius of the circle is six units long. We want to go ahead and write the inequality that shows the region inside the circle. So I'm just gonna go ahead and draw a quick picture. So we have this circle with the center located at negative three common negative four. We are told that the radius is six units long. And we want the inequality that shows the region inside of the circle. So how are we going to do this? Well, we can first start by writing out the general form of the equation of the circle not located at the center. That general equation is going to be X minus H squared plus Y minus K squared is equal to R squared. This is where the center is in the form of H comma K and R represents the radius. So we already know that the circle is not located at the center. And we are told that the center is going to be located at negative three, comma negative four. That means that H is going to equal to negative three and K is going to equal to negative four. And since we know that the radius is going to be six, that means that R squared is equal to six or I'm sorry, just R is equal to six. But what we need for our equation is R squared. So in order to get R squared, we can raise both sides of the equation to an exponent of two. And that is going to give us R squared is equal to six squared, which is 36. So now that we have R squared H and K, we can go ahead and plug this into the general form of the equation of the circle. That is going to give us X minus negative three squared plus Y minus negative four quantity squared is equal to R squared, which is 36. Now we're gonna go ahead and simplify and the X factor negative negative three is going to give us positive three. So we have X plus three squared and in the Y factor negative negative four is positive four. So we have Y plus four squared and that is going to equal to 36. Now keep in mind that we want the region inside of the circle. So we're going to have to represent this as an inequality. So we're gonna replace the, make the equal sign with an inequality. Now, here's the tricky part. What type of inequality sign do we use? Do we use greater than or equal to or do we use less than or equal to? While we're specifically talking about the region that is inside of the circle? So we want everything that's going to be less than the radius. So that means that we're going to replace the equal sign with less than or equal to. And this is going to be the inequality that represents the region inside the circle with the given center and radius. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Work each problem. Write the inequality that represents the region inside a circle with center (-5, -2) and radius 4.

Key Concepts

Circle Equation

Inequalities

Graphing Regions

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