Standard equation of a circle
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The equation of a circle C is x plus 3 squared plus y minus 4 squared is equal to 49. What are its 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 equidistant from that point. So let's take the set of all points that are, say, r away from h, k. So let's say that this distance right over here is r, and so we want all of the set of points that are exactly r away. So all the points x comma y that are exactly r way. 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 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 r away, at least if I were to draw it properly. They are r 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. 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, we're at y, here we're k, so this distance is going to be y minus k. We can do 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 minus h is this distance. And this is a right triangle, because by definition, we're saying, hey, 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 minus h squared plus y minus k squared must be equal to r squared. This is the equation for the set-- this describes any x and 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, it has to be in the form x minus h, y minus 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 that our h is negative 3-- I want to do that 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-- you know you might say, hey, there's a negative 4 here, no. But look, it's minus k, minus 4. So k is 4. Likewise, it's minus h. You might say, hey, maybe h is a positive 3, but no you're subtracting the h. So you'd say minus negative 3, and similarly, the radius is 7.