Distance and circles

Now, let’s go back to our plane, namely our plane equipped with our Cartesian coordinate system. And let’s consider two points in the plane. We’re now going to introduce the crucial concept of distance, which has been lacking so far. How do we define the distance between these two points? Well, the easiest way is to consider the segment PQ and imagine how we could calculate its length. The horizontal distance between P and Q, just the absolute value of the difference of the coordinates, is easily seen to be the base of a certain triangle whose vertical height is the absolute value of the difference of the y-coordinates.

If we now invoke the classical theorem of Pythagoras, you know, hypotenuse squared equals the sum of the squares of the other two sides, we then see that the distance P, Q, the length of that hypotenuse, will be the square root of x1 minus x2 squared plus y1 minus y2 squared. That’s our definition. It’s the Euclidean distance, as it’s called, between the two points P and Q. It has a lot of applications. Let’s consider a point C in the plane. And can we imagine a point that would be a positive distance, let’s say, r from the point C? So r is a fixed number here. And I want one point that is distance r from C. OK? There it is.

But there are others, obviously. I could go up a bit and find another point of distance r. Or I could move more to the left and find a third point of distance r. In fact, if I look at all the points I can find that are distance r from the point C, the locus of those points, the set of those points, defines what is called a circle, the circle of radius r and centered at the point C. Now, consider a point P on that circle. It has coordinates. Let’s call them x, y. I also need a name for the coordinates of the center of the circle. Let’s call it x0, y0. Why nought?

Well, actually nought, N-O-U-G-H-T is just a synonym for zero. So sometimes you would say x0, y0, same thing. What is the distance between P and C? Answer according to the formula we’ve have just determined, it’s what’s written here. And therefore, we can see that the point P, whose coordinates are x, y, lies on our circle if and only if it satisfies or its coordinates satisfy a certain equation for x and y. The equation that you see. This is called the general equation of a circle. It’s a circle centered at the point x0, y0 and of radius r greater than 0.

Now, if you were to expand the two squares of parentheses that you see in the general equation, and of course, you’d have a term in x squared and you’d have a term in y squared. And you notice that these terms will appear. And you’ll also have linear terms– x multiplied by a certain coefficient in front. And a y multiplied by a certain number in front of that and then another number. But you will have no x y. There’ll be no term in which the product x y occurs. We’ll see them later on in another connection. So when you see an equation of this type, it might be the expanded equation of a circle. You would expect it to be so.

Here’s an example in which we have to reverse the process of the expansion we’ve just seen and get back to the general equation. Suppose someone gives you this equation for the coordinates x, y of points in the plane. What does it describe? Well, you would anticipate a circle. You would anticipate a circle because the x squared is multiplied by 4, which is the same as what the y squared is multiplied by. And there’s no x y in the equation. How are you going to find the equation? Well, first, I divide it across by 4 just to have x squared and y squared without the coefficient 4. Next, you look at the terms involving x.

There are two of them– the x squared and the minus 4 x. And you’re going to do the process we have called in the past completing the square. That is, you’re going to write those two terms as x minus 2 squared minus 4, which is the same thing. This doesn’t really change anything. They’re just written in a way that will be more convenient. Similarly, you look at the two terms involving y– y squared plus 2 y. You complete the square with those two terms. And they can be written as y plus 1 squared minus 1. Now, you keep the 29/4. And you simplify a little bit.

And you’ll see that this equation, then, can be rewritten in the form x minus 2 squared, a positive term, plus y plus 1 squared, another positive term, equals something strictly negative. Well, of course, that’s crazy. You can’t have the sum of two positive things being negative. And in other words, what you have shown in this analysis, because the result here is negative on the right, is that this is not the equation of a circle. There are no points x, y that satisfy this equation. It’s what we will call the degenerate case of the equation of a circle. Didn’t really lead to a circle. Let’s look at another example, which may turn out differently.

Again, we see that in this equation there’s no x y term. And the coefficient of x squared, 3, is the same as the coefficient of y squared 3, so we expect a circle or maybe a degenerate case. How do we analyze? The same way. We group the terms with x and complete the square. We look at the two terms involving y. We complete the square with them. And we simplify the equation. And it becomes this. And now, we recognize the general equation of a circle, mainly because at the end on the right hand side you have something that can be r squared. It’s a circle. And what is the centre of the circle? The x0, y0.

You can read it off. It’s minus 1/3 minus 1. And what is the radius of the circle? Well, on the right, you have r squared. So the radius of the circle is root 2 over 3.

General equation of a circle. Remark– the circle is the level curve of a certain function. The function x minus x0 squared plus y minus y0 squared minus r squared. It’s a level curve of that function. But contrary to the graph of a non-vertical line, this does not correspond to the graph of a function y equals f of x. However, if you were to just restrict attention to those y’s that are greater than the value y0, then it does look like the graph of a function. And locally, we could solve for y in terms of x as indicated.

Graphically, it means that if you just look at the upper semicircle, then the x will go from x0 minus r to x0 plus r. And you know what the y is that corresponds for any x in that interval. The y can be solved for to get this. So a piece of our level curve, in this case the upper semicircle, is the graph of a function.

We’ll be seeing a certain number of other famous level curves in the following segments.