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Circles in the x-y Plane


In the previous video, I suggested that a circle centered at the origin with a radius of r would have the formula, x squared plus y squared equals r squared. We concluded this by thinking about distance in the x, y plane and in particular thinking about the application of the Pythagorean theorem to the coordinate plane. Now we can ask, what if the origin is not the center of the circle?

What is the equation then, if we have a circle centered somewhere else in the coordinate plane? Well, let's think about this. We'll have a circle centered at some general point, the center is (h, k), has a radius of r, and we want to know the equation of that circle. Well look at that little right triangle.

The right triangle that has a hypotenuse of r. It has a horizontal leg and a vertical leg. The horizontal leg is this distance, right here, between x and h. So it would be (x-h) or at least the absolute value of (x-h). Similarly, the vertical leg is this distance, right here, between y and k, so y minus k.

Or the absolute value of that. It doesn't matter cause we're going to square those anyway. And so the horizontal leg squared plus the vertical leg squared must equal the hypotenuse squared. Of course, the hypotenuse is r, and so that gives us the formula (x-h) squared plus (y-k) squared equals r squared, and that is the general formula for a circle in the coordinate plane with center h, k and radius r.

Once again, I will encourage you not simply to memorize that formula, but to understand the logic that underlies that formula. You will understand it much more deeply if you understand where that formula comes from. If the radius is r and the center is (h,k), then the equation of the circle is (x-h) squared (y-k) squared equals r squared.

Here's a practice problem. Pause the video and then we'll talk about this. Okay, a square in the x-y plane has the vertices (-1,1), (3,1), (3, -3) and (-1, -3). A circle is tangent to all four sides. What is the equation of the circle?

So they're very nice to give us a diagram we can see this. Well, first of all, notice the radius of this circle is 2. So, right away we know that the r squared side has to be 4. So all these ones that have 9 can't be correct. We know the center is at (1,-1). The x coordinate of the center is a positive number.

The y coordinate of the center is a negative number. As we can see from the diagram, the center is in the fourth quadrant. And so this means that when we subtract, we're gonna have (x- 1), but for the y, we're gonna have (y- -1). Well minus negative 1 is plus 1, so it's gonna be x-1 and y+1, those are gonna be the two components of it, and so that has to be answer choice B.

Here's another question, a bit of an odd question. Pause the video and then we'll talk about this. So it gives us the equation of a circle, a square is drawn around the circle and each side is tangent to the circle. For each vertex of this square, we compute the sum of the x and y coordinates. The highest of the four sums is what?

It's a very unusual question. So let's think about this. First of all, from the equation we know the center is (-4, 2). Very important, because it's x+4. That means that the x coordinate of the center is -4, y-2, that is the center at positive 2.

And of course the radius is 5. Well, from the x coordinate of x=-4, if we go to the left five spaces, we get to x=-9. We go to the right five spaces, we get to x=+1. From the y coordinate of 2, we go down five spaces, we get to the line y.=-3, and if we go up five spaces, we get to the line y =-7.

So technically, those four lines, those two horizontal lines to those two vertical lines define the square. And so this means that the vertices would be the places where those lines intersect. So the vertices are these four points. And we take the sum, we see that the highest sum is the one with the two positive numbers, 1 and 7.

1+7=8, so that's the highest sum. And it kinda makes sense that the highest sum would be the one that would be on the upper right side of the square, that the vertex on the upper right side obviously is gonna have larger coordinates than any of the other three coordinates. So, the sum is 8. We go back to our choices and we choose 8.

In summary, if the radius is r and the center is (h,k) then the equation of the circle is (x-h) squared plus (y-k) squared equals r squared. Once again, I would urge you, do not simply memorize this by rote, but understand the argument that produced this equation. That will make you much more confident in using it.

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