In this video, we discuss and illustrate lines and circles in the Cartesian plane and their associated equations. Now, why lines and circles? Well, they are amongst the most fundamental classical objects in geometry, and they turn out to be of fundamental importance in calculus. Lines involves simple arithmetic. But isn't calculus meant to be complicated? No. Calculus is the art of thinking deeply about simple things, and lines are those simple things we need. As you will see, lines are the core business of calculus, which aims to reduce complicated processes to simple arithmetic. What about circles? Well, they model repetitive or periodic behavior. You've heard the expression "going around in circles" which sounds like you're not getting anywhere. Well, repeated patterns are part of our life. Think of day and night, the tides, the seasons, your heartbeat. Where would we be without our hearts beating? The mathematics of circles is fundamental to understanding phenomena that repeat indefinitely. Let's begin with lines in the plane. They have general equations in the form ax plus by equals c, where x and y are variables, and a, b and c are constants. You might be familiar with other forms, but this one is probably the most used by mathematicians because it can be easily generalized as I might explain later. For example, let's sketch the line 2x plus 3y equals 6 where a is equal to 2, b is equal to 3, and c is equal to 6. Here's a copy of the xy plane. If we can find two points on the line, then we just need to join them up, and it's simplest to look for points on the axes. When x is equal to 0, the equation becomes 3y equals 6, so y is equal to 2. So the y-intercept is 2, and we can mark off 2 on the y-axis. Now, when y is equal to 0, the equation becomes 2x is equal to 6, so x is equal to 3, and we can mark off 3 on the x-axis and then join the points to create a line. Now, we'll just use the general principle that a line is determined by two points. In general, we call these points say P and Q and draw a line between them. Notice that we have horizontal and vertical directions in the xy plane and this leads to the notion of slope, which is the fraction or proportion obtained by dividing the vertical rise by the horizontal run, typically denoted by m. Now, if the horizontal run is just one unit, then we're dividing by 1 so the fraction becomes the vertical rise. In other words, we move m units vertically for each unit we move horizontally, and it turns out there's a nice formula for m in terms of the coordinates of P and Q. Here are P and Q again, but now with coordinates say, x1, y1 for P and x2, y2 for Q. I've added a little right-angled triangle with horizontal side length 1 and vertical side length m. Now, let's build a large right-angled triangle that captures movement from P to Q but resolved into horizontal and vertical components. The horizontal run is just x2 minus x1 in this diagram whereas the vertical rise is y2 minus y1. Can you see two similar right-angled triangles? Because the two triangles we've highlighted are similar, the ratios of yellow to green are the same. For the smaller triangle, this is m divided by 1. For the larger triangle, this is y2 minus y1 divided by x2 minus x1. Thus, we've established that the slope m is the ratio of the differences in the coordinates, and we get this nice formula. It doesn't matter if you reverse the roles of P and Q as the ratio stays the same as negatives cancel in the numerator and denominator. So, for example, what's the slope of the line joining P and Q with coordinates 2,2 and 5,6? We can just apply the formula to get m equal to 6 minus 2 divided by 5 minus 2, which is 4 over 3, or if you reverse the order of the points, you get m is 2 minus 6 on 2 minus 5. That's minus 4 and minus 3, which still evaluates to four-thirds. How might we visualize this? Here's a copy of the xy plane and add the points P and Q, and draw a line between them. As you move from P to Q, you can see that the horizontal run has three units, and the vertical rise is four units. So the slope is four-thirds a bit more than 1. So, the angle of inclination is a bit more than 45 degrees, and we'll have a lot more to say about angles in the next module. Now, lines can have positive, negative or zero slope. Depending on whether the line slopes upwards, downwards or are horizontal as you move from left to right. Actually, there's one more possibility, infinite slope, where you don't move at all left or right, which will come to you shortly. Here's some more practice with points P and Q with mixtures of positive and negative coordinates. Applying the formula for the slope, we quickly get that m is equal to minus 1. Visually, we can say that when we join the points, the line slopes downward with slope minus 1. One unit downwards vertically for one unit horizontally from left to right, and this is very clear, say, moving from the y-intercept to the x-intercept. But how do we find the equation of a line? Well, with the general form ax plus by equals c can be transformed in a few steps into the form y equals mx plus k, where m is equal to minus a on b and k is equal to c on b. Provided the constant b is non-zero, m is our old friend, the slope, and k turns out to be the y-intercept, where x is equal to 0. The line passes through the y-axis at y equals k and moves m units vertically for each unit moved horizontally. Here's some practice. Find the equation of the line passing through P and Q with coordinates 2,3 and 7,13. To solve this, we must first find the slope m and quickly see that it evaluates to two. So, the equation must be the form 2x plus k for some k. We can find k by subbing in coordinates of one of the points, say P, and you quickly find that k is equal to minus 1. Thus, the equation is y equals 2x minus 1. You can quickly check that this works for P, of course, but also as a double check for Q. Here is the graph passing through the y-axis at minus one, and as you move one unit across to the right, the line moves up two units vertically. Warning. Some lines are vertical. So, have what you might think of it as an infinite slope and do not have the form y equals mx plus k. In our derivation of y equals mx plus k before, we assumed that b is not equal to 0. So, what happens if b is equal to 0? The equation becomes ax equals c. Now dividing by a, we get that x is equal to, say l, where l is the constant c divided by a. Note that if a and b are both equal to 0, then c is equal to 0 and the equation is degenerate and not very interesting. So, assuming b is equal to 0, it forces a to be nonzero. Here is what the vertical line x equals l looks like passing through the x-axis at l. So, why do we emphasize the general form ax plus by equals c? The reason is that it generalizes. Remember how Descartes went from one to two to three copies of the real line forming axes? If we add a third variable, say z, then there is a natural pattern, and the new equation should become ax plus by plus cz equals d, where now a, b, c, and d are constants. Now, your first guess might be that this is the equation of a line in space, but in fact, it is the equation of a two dimensional plane in space. Like the walls, floor, or ceiling in your room, each is a plane in space described by an equation like that. Why two-dimensions? Though there are three variables x, y, and z, the equation places a constraint. So that after choosing any two of the variables independently, the third is completely determined by the equation in algebraic manipulation. We say that there are two degrees of freedom. Why stop at three variables? Well, further generalizations are mentioned in the notes if you're interested, and you'll become very familiar with higher dimensions if you do more mathematics. Now, returning to the xy-plane, let's discuss circles and how to describe them. This is where Pythagoras comes in. Consider a circle centered at P with radius r, what should its equation look like? Well, let Q be a typical point on the circle with coordinates, say x and y, and you can imagine Q spinning around the circle. We want to understand the relationships between x, y, a, b, and r. So, we form a right-angled triangle involving P and Q, using horizontal and vertical side lengths. In this diagram, the horizontal side length is x minus a and the vertical side length is y minus b. By Pythagoras, the square of the hypotenuse is the sum of the squares of the sides of the triangle. So, r squared is x minus a squared plus y minus b squared. That gives us the equation of the circle, though we usually write it around the other way with r squared on the right. Let's do an example. Say, the circle C centered at P with coordinates 3,4 and radius 5 units. We can simply read off the equation, where a is equal to 3, b is equal to 4 and r is equal to 5, and you could ask various questions, like whether the circle crosses the axes, and if so, where? Let's look at the y-axis, where x is equal to 0. So, what does this imply? Both the equation simplifies. In several steps, you see that y is equal to 0 or 8. So, there are two answers, points on the y-axis with coordinates 0,0 and 0,8. Now, what about crossing the x-axis, where y is equal to 0? Again, the equation simplifies. In a few steps, we see that x is equal to 0 or x is equal to 6. So, the points on the x-axis have coordinates 0,0 and 6,0. Notice that the origin, the intersection of the x and y-axis appears both times. Here, the picture tells the full story for this particular circle. We chose this example so the numbers fall out easily, in fact, using a 3, 4, 5 right-angled triangle. But in principle, you can use the equation of a circle to find out lots of information, regardless of what the constants are. Typically, these calculations involve square roots. We've covered a lot of ground, equations of lines and circles, gradients and intercepts with axes, and an application to circles with the Theorem of Pythagoras. The important idea is that the geometry of lines and circles can be captured algebraically by these equations. Please read the notes and when you're ready, please attempt the exercises. Now, this is the last video in the first module, and you may want to review ideas and techniques introduced so far. Let's see. We've talked about the real number line and decimal expansions, the Theorem of Pythagoras in an historical context, and properties of the square root of 2. We've talked about surd expressions and some novel applications of algebraic manipulation, finding solutions to equations and inequalities, where we've exploited sign diagrams and interval notation. We have discussed coordinate systems, in particular, Descartes' wonderful invention of the xy-plane, distance and absolute value, and now today, lines and circles in the plane. Thank you very much for your participation, and I look forward to seeing you again soon.
