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CCSS.Math:

the equation of a circle c is x plus 3 squared plus y minus 4 squared is equal to 49 what are at center H K and its radius R so let's just remind ourselves what a circle is you have some point let's call that H K the circle is the set of all points that are that are equidistant from that point so let's take the set of all points that are say R away from HK so let's say that this distance right over here this distance right over here is R and so we want all the set of points that are exactly R away so all the points X comma Y that are exactly R away and so you could imagine you could rotate around and all of these points are going to be exactly R away and I'm going to try my best to to draw to draw at least a somewhat perfect looking circle I won't be able to do a perfect job of it but you get a sense all of these are exactly they're exactly R away at least if I were to draw it properly there are away so how do we find an equation in terms of R and H K and x and y that describes all these points well we know how to find the distance between two points on a coordinate plane and in fact in fact it comes straight out of the Pythagorean theorem if we were to draw a vertical line right over here that essentially is the change in the vertical axis between these two points up here where at Y here where K so this distance is going to be Y minus K we can do the exact same thing on the horizontal axis the exact same thing on the horizontal axis this x coordinate is X while this x coordinate is H so this is going to be X this is going to be X minus H is this distance and this is a right triangle because we by definition we're saying hey this is a vertical we're measuring vertical distance here we're measuring horizontal distance here so these two things are perpendicular and so from the Pythagorean theorem we know that this squared plus this squared must be equal to our distance squared and this is where the distance formula comes from so we know that X - h squared plus plus y minus K squared must be equal to R squared this is the equation for the set this describes any X&Y that satisfies this equation will sit on this circle now with that out of the way let's go answer their question the equation of the circle is this thing and this looks awfully close to what we just wrote we just have to make sure that we don't get confused with the negatives remember has to be in the form X - H Y - K so let's write it a little bit differently instead of X plus 3 squared we can write that as X minus negative 3 squared and then plus well this is already in the form plus y minus 4 squared is equal to instead of 49 we can just call that 7 squared and so now it becomes pretty clear pretty clear that our H is negative 3 I wanted to do that in a I want to do that in it in the red color that our H is negative 3 and that our K is positive 4 and that our R is 7 so we could say H comma K is equal to negative 3 comma positive 4 make sure to get the you know you might say hey there's a negative 4 you know but look it's minus K minus 4 so K is 4 likewise it's minus H you might say hey maybe HS a positive 3 but no it's you're subtracting the H so you say minus negative 3 and similarly the radius is 7