WEBVTT 00:00:02.440 --> 00:00:07.913 This particular video is going to be about of a set of curves 00:00:07.913 --> 00:00:09.597 known as conic sections. 00:00:10.290 --> 00:00:12.540 Here's a cone. 00:00:13.080 --> 00:00:19.151 And conic sections are formed by taking cuts in into this cone in 00:00:19.151 --> 00:00:23.830 various ways. Now, the curves that are formed by taking these 00:00:23.830 --> 00:00:28.030 particular cuts or sections are very old. They were known to the 00:00:28.030 --> 00:00:29.780 ancient Greeks from about 3:50 00:00:29.780 --> 00:00:35.244 BC onwards. But not only they very old, but they also a very 00:00:35.244 --> 00:00:39.709 modern. The dish aerial that's outside your house that brings 00:00:39.709 --> 00:00:44.010 in Sky Television is based upon the reflective property of one 00:00:44.010 --> 00:00:48.702 of these conic sections that we're going to be having a look 00:00:48.702 --> 00:00:52.221 at Georgia Bank Telescope, a radar telescope that pierces 00:00:52.221 --> 00:00:56.131 deep into the universe, picking up signals, recording them for 00:00:56.131 --> 00:01:00.432 us again, is based upon a reflective property of one of 00:01:00.432 --> 00:01:03.951 these conic sections. So although they are ancient, even 00:01:03.951 --> 00:01:06.297 though they are, they have very, 00:01:06.297 --> 00:01:08.418 very. Important modern uses. 00:01:09.070 --> 00:01:13.246 So let's have a look at our coal. Let's begin by describing 00:01:13.246 --> 00:01:17.074 some features about the cone itself, which I'll need in order 00:01:17.074 --> 00:01:21.250 to describe the sections and the curves that we're going to see. 00:01:21.250 --> 00:01:25.426 First of all, let's think about the axis of the cone. Now, 00:01:25.426 --> 00:01:29.254 that's the line that runs from the point down through the 00:01:29.254 --> 00:01:32.734 center of the basin out there. So that's the axis. 00:01:33.360 --> 00:01:37.824 The generator of the comb. Well, that's any line that runs down 00:01:37.824 --> 00:01:42.660 the side of the cone like that and the generator. Think of it 00:01:42.660 --> 00:01:47.124 as being a pendulum, a piece of string tight there. Its wings 00:01:47.124 --> 00:01:51.588 round here and it Maps out this curved surface of the cone. 00:01:52.480 --> 00:01:57.745 So let's begin by looking at the cuts or sections that we can 00:01:57.745 --> 00:02:02.606 make. One of the simplest sections that we can make is to 00:02:02.606 --> 00:02:06.878 cut across and perpendicular to the axis of the cone. And what 00:02:06.878 --> 00:02:10.438 do we get? We get that well known curve circle. 00:02:11.550 --> 00:02:17.100 OK, let's not now cut perpendicular. Let's cut at an 00:02:17.100 --> 00:02:20.960 angle. So we're going to cut at an angle to the. 00:02:21.780 --> 00:02:23.468 Access of the cone. 00:02:23.990 --> 00:02:27.640 We're going to keep the cut within the physical dimensions 00:02:27.640 --> 00:02:32.385 of the cone. And what do we get? We get a curved surface. 00:02:33.120 --> 00:02:37.509 A closed curve again, this time it's an ellipse, kind of 00:02:37.509 --> 00:02:41.898 flattened circle if you like, but its name is an ellipse. 00:02:42.800 --> 00:02:44.210 Let's put that back on. 00:02:44.790 --> 00:02:51.126 And now let's take a cut which is parallel till the 00:02:51.126 --> 00:02:54.006 generator parallel to the edge 00:02:54.006 --> 00:02:59.386 of the. Curved surface, so there's our edge. 00:03:00.400 --> 00:03:05.639 Of the curve surface, and this is the cut that we're going to 00:03:05.639 --> 00:03:09.669 take parallel to that edge. So we make the cut. 00:03:10.300 --> 00:03:14.452 And there we've got an open curve, this time, not a closed 00:03:14.452 --> 00:03:17.912 one. It's an open curve and it's called the parabola. 00:03:18.640 --> 00:03:23.957 Putting that back together, what I want you to imagine now is a 00:03:23.957 --> 00:03:29.683 double Cole. Now what do I mean by a double cone? I mean another 00:03:29.683 --> 00:03:32.546 cone sitting on top of this one. 00:03:33.430 --> 00:03:38.770 Point to point. So imagine we've got this cone here. We've got 00:03:38.770 --> 00:03:40.550 another cone up here. 00:03:41.110 --> 00:03:46.318 And again, I'm going to take a cut, which this time is going to 00:03:46.318 --> 00:03:50.782 be parallel to the axis of the coal coming straight down there, 00:03:50.782 --> 00:03:55.990 and you can see the cut there now. Once I make it there, I've 00:03:55.990 --> 00:04:00.454 made the cut will actually be making the cut through the cone 00:04:00.454 --> 00:04:04.918 up here, and so they'll actually be 2 pieces to the curve. 00:04:06.030 --> 00:04:11.646 One from the upper cone and one from the lower cone, and that's 00:04:11.646 --> 00:04:15.102 a hyperbola. It's got two branches to it. 00:04:15.850 --> 00:04:20.512 So those are our three conic sections, sorry, miscounted. 00:04:20.512 --> 00:04:24.656 Those are our four conic sections, the circle. 00:04:26.390 --> 00:04:27.440 The ellipse. 00:04:28.780 --> 00:04:35.578 The parabola. On the hyperbola with 00:04:35.578 --> 00:04:38.254 its two branches. 00:04:38.820 --> 00:04:42.384 Now the conic sections were first discovered by monarchists 00:04:42.384 --> 00:04:47.532 in about 350 BC. Minarchist was an ancient Greek. He in fact had 00:04:47.532 --> 00:04:52.284 three different cones, but he took the same cut in each cone 00:04:52.284 --> 00:04:54.660 in order to generate three of 00:04:54.660 --> 00:04:58.194 them. Here with disregarding the circle, 'cause, That's obviously 00:04:58.194 --> 00:04:59.809 the simplest cut to make. 00:05:00.540 --> 00:05:05.810 Later Apollonius who was around about 200 BC, generated this 00:05:05.810 --> 00:05:11.607 idea of making the cuts all in exactly the same cone. 00:05:12.340 --> 00:05:17.715 So. We've seen what these curves look like, and we've 00:05:17.715 --> 00:05:23.110 seen how they are generated now. What we want to do is to 00:05:23.110 --> 00:05:24.770 explore their mathematical equations. 00:05:25.790 --> 00:05:28.913 If we don't explore these mathematical equations, then we 00:05:28.913 --> 00:05:33.424 won't be able to deal with these curves and find out what their 00:05:33.424 --> 00:05:37.720 properties are. So let's have a look at a definition of these 00:05:37.720 --> 00:05:40.474 curves that will actually help us to calculate them. 00:05:41.010 --> 00:05:45.300 What I want first of all is a straight line. Straight line has 00:05:45.300 --> 00:05:50.020 a name. It's called the direct tricks. 00:05:51.060 --> 00:05:58.366 And I want a fixed point. I'm going to label that fixed point 00:05:58.366 --> 00:06:01.176 F and it's F4 focus. 00:06:03.200 --> 00:06:05.780 Take any point. 00:06:06.490 --> 00:06:10.423 In the plane, let's call that the point P. 00:06:11.600 --> 00:06:17.723 And then we've got P must be a distance away from this fix 00:06:17.723 --> 00:06:23.375 straight line. Let's call that point M and that's of course a 00:06:23.375 --> 00:06:29.027 right angle. And it's also a distance away from F. The focus. 00:06:29.600 --> 00:06:35.324 And so the definition that we're going to use is that PF. 00:06:35.870 --> 00:06:39.176 The distance of the point from 00:06:39.176 --> 00:06:42.909 the focus. Is equal to. 00:06:43.460 --> 00:06:48.652 A constant multiple of the distance of the point from the 00:06:48.652 --> 00:06:54.316 straight line and that constant multiple. We use the letter E to 00:06:54.316 --> 00:07:00.390 denote. And so PF is equal to E times by PM. 00:07:01.370 --> 00:07:06.698 Now what happens? Well, this point P follows what we call a 00:07:06.698 --> 00:07:12.026 locus. Let's just write down that word locus. What do we mean 00:07:12.026 --> 00:07:18.242 by a locus? Well, it's simply a path. The Point P follows a path 00:07:18.242 --> 00:07:20.018 governed by this rule. 00:07:21.100 --> 00:07:28.540 This letter EE is used to stand for what we 00:07:28.540 --> 00:07:30.772 call the eccentricity. 00:07:31.600 --> 00:07:37.536 Of the curve and we get our different. 00:07:38.080 --> 00:07:43.976 Conic sections for different values of E. So if E is 00:07:43.976 --> 00:07:50.944 between North and won, the curve that we get is the closed curve. 00:07:51.640 --> 00:07:53.060 The ellipse. 00:07:54.100 --> 00:07:58.940 If E is equal to 1, the curve that we get. 00:07:59.740 --> 00:08:01.210 Is the parabola. 00:08:01.830 --> 00:08:04.756 The open curve but with a single 00:08:04.756 --> 00:08:11.246 branch. And if he is greater than one, the curve that we get 00:08:11.246 --> 00:08:16.622 is the hyperbola. Remember that was the open curve with the two 00:08:16.622 --> 00:08:21.258 branches. Now what happens in the rest of the video? 00:08:21.258 --> 00:08:24.154 We're going to deal with this particular curve. 00:08:26.060 --> 00:08:31.182 In some detail we look at some of its properties and will just 00:08:31.182 --> 00:08:35.122 relat say what the equivalent properties are in these two 00:08:35.122 --> 00:08:40.244 curves. But this is the one that we're going to deal with in 00:08:40.244 --> 00:08:44.578 detail, and the sorts of mathematics that we use and the 00:08:44.578 --> 00:08:48.912 ways in which we work if replicated will enable the same 00:08:48.912 --> 00:08:53.246 sorts of properties to be proved and established in the ellipse 00:08:53.246 --> 00:08:55.610 and the hyperbola. So let's go 00:08:55.610 --> 00:08:58.218 on. Have a look at the parabola. 00:08:59.520 --> 00:09:02.288 So. Let's draw a picture. 00:09:04.820 --> 00:09:09.960 So I'm going to set up some coordinate axes X&Y. 00:09:10.650 --> 00:09:14.410 I'm going to take my straight line my direct tricks. 00:09:15.300 --> 00:09:18.028 There, and I'm going to take my focus. 00:09:19.050 --> 00:09:25.291 There. Now I need some measure, some scale, some size, so I'm 00:09:25.291 --> 00:09:30.625 going to say I'm going to take the focus at the point a nought. 00:09:31.910 --> 00:09:37.520 Now remember that 4 hour parabola. 00:09:38.540 --> 00:09:45.570 PF is equal to PM be'cause E the eccentricity was 00:09:45.570 --> 00:09:47.679 equal to 1. 00:09:48.310 --> 00:09:52.899 So the distance of our point P has to be the same from. 00:09:53.670 --> 00:09:59.180 This line. As it is from this point and of course, there's an 00:09:59.180 --> 00:10:02.240 obvious point, namely the origin, that perhaps we'd like 00:10:02.240 --> 00:10:06.320 to be on this curve. So that means that the direct tricks 00:10:06.320 --> 00:10:09.040 here is going to be at X equals 00:10:09.040 --> 00:10:12.108 minus a. Let's put our. 00:10:12.890 --> 00:10:16.060 Point P. Let's say there. 00:10:17.380 --> 00:10:18.720 There is. 00:10:19.970 --> 00:10:27.076 PM That's our point. N there is PF 00:10:27.076 --> 00:10:31.684 will call this the point XY. 00:10:33.280 --> 00:10:38.128 So what's the locus? What path? What's the equation of that path 00:10:38.128 --> 00:10:42.976 that P is going to follow as it moves according to this 00:10:42.976 --> 00:10:47.824 definition, one of the things we better do is we better write 00:10:47.824 --> 00:10:49.844 down what are these lens? 00:10:50.600 --> 00:10:57.838 PM equals well from M to the Y axis is a distance A and 00:10:57.838 --> 00:11:04.559 there's a further distance of X to go before we get to pee. 00:11:04.559 --> 00:11:07.661 So PM is A plus X. 00:11:08.340 --> 00:11:11.790 What about PF? 00:11:12.900 --> 00:11:18.204 Once we look at PF, it's the hypotenuse of a right angle 00:11:18.204 --> 00:11:22.624 triangle which we can form by dropping a perpendicular down 00:11:22.624 --> 00:11:25.718 there so we can see that PF. 00:11:27.060 --> 00:11:31.656 Will be when we square it. PF squared will be equal to 00:11:31.656 --> 00:11:34.720 that squared, which is just the height, why? 00:11:35.860 --> 00:11:39.760 Plus that squared. 00:11:40.390 --> 00:11:47.615 Will up to there is X and up to there is a so that little bit in 00:11:47.615 --> 00:11:49.740 there is a minus X. 00:11:50.340 --> 00:11:51.360 Squared 00:11:54.620 --> 00:11:58.080 So this is PF squared. 00:11:58.670 --> 00:12:03.734 Let's just check that again, PF is the hypotenuse of a right 00:12:03.734 --> 00:12:07.532 angle triangle, so using Pythagoras PF squared is that 00:12:07.532 --> 00:12:11.752 squared plus that squared. That's the height YP is above 00:12:11.752 --> 00:12:13.018 the X axis. 00:12:13.640 --> 00:12:19.880 Up to there is X up to. There is a, so the distance between there 00:12:19.880 --> 00:12:22.376 and there is a minus X. 00:12:22.950 --> 00:12:28.943 So now I need to equate these two expressions and first of all 00:12:28.943 --> 00:12:35.397 that means I've got to square PM because if PF is equal to PMPF 00:12:35.397 --> 00:12:41.964 squared. Must be equal to PM squared and so we can substitute 00:12:41.964 --> 00:12:49.580 this in so instead of PF we have Y squared A minus, X all 00:12:49.580 --> 00:12:56.108 squared and instead of PM squared we have a plus X all 00:12:56.108 --> 00:13:02.732 squared. Now we need to multiply out this bracket Y 00:13:02.732 --> 00:13:08.414 squared plus. Now let's just do the multiplication over here. 00:13:08.920 --> 00:13:15.318 A minor sex, all squared is a minus X times by A minus X. 00:13:15.890 --> 00:13:20.587 So we've a Times by a that gives us a squared. 00:13:21.330 --> 00:13:27.210 With a Times by minus X and minus X times by a, which gives 00:13:27.210 --> 00:13:33.510 us minus two AX and then we have minus X times Y minus X which 00:13:33.510 --> 00:13:38.550 gives us plus X squared equals. Let's have a look at this 00:13:38.550 --> 00:13:43.590 bracket. It's a plus X all squared, so it's going to be 00:13:43.590 --> 00:13:48.630 exactly the same as this one, except with plus signs in. So 00:13:48.630 --> 00:13:51.570 we're going to get a squared +2. 00:13:51.600 --> 00:13:55.100 X plus X squared. 00:13:56.360 --> 00:14:00.980 Now let's have a look at this. We've A plus a squared here and 00:14:00.980 --> 00:14:05.930 a plus a squared there so we can take an A squared away from each 00:14:05.930 --> 00:14:10.220 side. We have a plus X squared here under plus X squared here 00:14:10.220 --> 00:14:12.860 so we can take an X squared away 00:14:12.860 --> 00:14:19.276 from each side. We've minus two X here and plus 2X there, so it 00:14:19.276 --> 00:14:24.508 makes sense to get those axis together by adding 2X to this 00:14:24.508 --> 00:14:27.560 side and adding it to that side. 00:14:28.070 --> 00:14:31.710 So if we do all that we've got Y squared. 00:14:33.140 --> 00:14:37.901 The two A squared will disappear, subtracting a square 00:14:37.901 --> 00:14:42.662 from each side. The two X squared will disappear. 00:14:42.662 --> 00:14:49.010 Subtracting X squared from each side and adding the two X2 each 00:14:49.010 --> 00:14:54.829 side, we get 4A X that is the standard Cartesian equation 00:14:54.829 --> 00:14:56.416 for a parabola. 00:14:57.500 --> 00:14:59.040 Now. 00:15:00.400 --> 00:15:04.274 This is a standard equation and it is the equation with which we 00:15:04.274 --> 00:15:08.826 want to work. But sometimes when we're doing these 00:15:08.826 --> 00:15:13.560 curves, it's helpful to have the equation described in 00:15:13.560 --> 00:15:15.664 terms of 1/3 variable. 00:15:17.110 --> 00:15:22.330 Variable that's often called a parameter. 00:15:23.480 --> 00:15:28.100 And the parameter we're going to have in this case is the 00:15:28.100 --> 00:15:29.630 parameter. T. 00:15:30.840 --> 00:15:37.880 So we've got an equation Y squared equals 4A X. 00:15:38.680 --> 00:15:43.650 And what we're looking for is a way of expressing X in terms of 00:15:43.650 --> 00:15:47.555 tea and a way of expressing Y in terms of T. 00:15:48.590 --> 00:15:54.372 When we look at this, this says Y squared equals 4A X. Now Y 00:15:54.372 --> 00:15:58.915 squared is a complete square. It's a whole square an exact 00:15:58.915 --> 00:16:03.871 square. I can take it square root and I'll just get plus 00:16:03.871 --> 00:16:09.653 online as why so? Can I do it over here? Can I choose an 00:16:09.653 --> 00:16:13.783 expression for X that will give me a complete square? 00:16:14.480 --> 00:16:18.560 On this side of the equation, well, four is already a complete 00:16:18.560 --> 00:16:23.320 square. If I want to make a a complete square and have to have 00:16:23.320 --> 00:16:27.740 a squared. So I need something in X that's gotten a attached to 00:16:27.740 --> 00:16:31.480 it. So 8 times by a would give me a squared. 00:16:32.070 --> 00:16:36.295 And then I want to be able to take this exact square root. 00:16:36.295 --> 00:16:39.545 Will the only thing if I'm going to introduce this 00:16:39.545 --> 00:16:42.795 variable T that suggests itself and he sensible is T 00:16:42.795 --> 00:16:44.420 squared? So if we put. 00:16:45.990 --> 00:16:52.510 X equals AT squared, then Y squared is 00:16:52.510 --> 00:16:59.030 equal to four a Times 80 squared, which 00:16:59.030 --> 00:17:05.550 is 4A squared T squared. And so why 00:17:05.550 --> 00:17:08.810 is equal to two 00:17:08.810 --> 00:17:14.350 AT? And So what I've got here is what's called 00:17:14.350 --> 00:17:15.670 the parametric equation. 00:17:16.830 --> 00:17:20.220 For the parabola. 00:17:22.100 --> 00:17:24.540 Now that's our parametric 00:17:24.540 --> 00:17:31.269 equation. And this is our Cartesian Equation. One thing we 00:17:31.269 --> 00:17:34.274 haven't done yet is sketched. 00:17:34.910 --> 00:17:40.022 The curve itself, so just want to do that. 00:17:41.890 --> 00:17:42.919 So there's a. 00:17:43.600 --> 00:17:45.580 X&Y axis. 00:17:46.830 --> 00:17:50.386 There's our direct tricks through minus a. 00:17:51.130 --> 00:17:54.718 Here's our focus at a nought. 00:17:55.930 --> 00:18:00.597 We know the curve is going to go through there and it's equation 00:18:00.597 --> 00:18:05.623 is Y squared equals 4A X. Well, if we take positive values of X, 00:18:05.623 --> 00:18:10.649 we can see we're going to get that, but a positive value of X 00:18:10.649 --> 00:18:15.675 gives us the value of Y, which when we take the square root is 00:18:15.675 --> 00:18:20.342 plus or minus. So we can see we're going to have that and. 00:18:20.870 --> 00:18:25.787 Symmetry in the X axis. Notice we can't have any negative 00:18:25.787 --> 00:18:31.598 values of X because 4A A is a positive number times a negative 00:18:31.598 --> 00:18:36.962 value of X would give us a negative value for Y squared. 00:18:37.810 --> 00:18:41.960 What about the parametric equation X equals 80 squared Y 00:18:41.960 --> 00:18:47.355 equals 280? What does this do in terms of this picture? Well, if 00:18:47.355 --> 00:18:53.165 we look at it, we can see X equals 80 squared. A is a 00:18:53.165 --> 00:18:57.315 positive number, T squared is positive, so again we've only 00:18:57.315 --> 00:19:02.295 got values of X which are greater than or equal to 0. 00:19:03.470 --> 00:19:09.630 However, values of why can range from minus Infinity to plus 00:19:09.630 --> 00:19:15.790 Infinity? And if you like tea down here is minus Infinity. 00:19:16.930 --> 00:19:22.910 Going round to T equals 0 here at the origin coming round. If 00:19:22.910 --> 00:19:26.130 you like to T is plus Infinity 00:19:26.130 --> 00:19:33.015 there. So T is a parameter and in a sense it counts is around 00:19:33.015 --> 00:19:37.800 the curve as T increases from minus Infinity to plus Infinity, 00:19:37.800 --> 00:19:39.975 we move around the curve. 00:19:40.490 --> 00:19:47.060 OK. What about this curve? Well, one of 00:19:47.060 --> 00:19:53.300 the things that we want to be able to establish is what's the 00:19:53.300 --> 00:19:58.580 equation of the tangent to the curve at a particular point. 00:20:00.260 --> 00:20:05.220 So again, let's draw our picture of the curve, putting 00:20:05.220 --> 00:20:10.180 in the direct tricks straight line, putting in the focus 00:20:10.180 --> 00:20:11.668 and putting in. 00:20:12.690 --> 00:20:13.670 The curve. 00:20:15.200 --> 00:20:20.680 Let's take any point P on the curve. 00:20:21.380 --> 00:20:27.373 And the question we are asking is what is the equation of the 00:20:27.373 --> 00:20:29.217 tangent to the curve? 00:20:30.500 --> 00:20:37.234 What's the equation of that lie? Well, I've got P is any point on 00:20:37.234 --> 00:20:43.006 the curve. So instead of calling XY, I'm going to use the 00:20:43.006 --> 00:20:49.276 parametric equation. X is 80 squared, Y is 280. 00:20:50.150 --> 00:20:56.855 So X is 80 squared, Y is 2 AT. 00:20:57.750 --> 00:21:01.996 The question we're asking what is the equation of the tangent? 00:21:01.996 --> 00:21:06.628 Well, it's a straight line to find the equation of a straight 00:21:06.628 --> 00:21:12.032 line, we need a point on that line or we've got it 80 squared 00:21:12.032 --> 00:21:17.436 280 and we need the gradient of the line and the gradient of the 00:21:17.436 --> 00:21:22.068 line. Of course must be the gradient of the curve at that 00:21:22.068 --> 00:21:25.156 point P, so we need DY by X. 00:21:26.680 --> 00:21:32.359 But both X&Y are defined in terms of T. 00:21:33.130 --> 00:21:38.200 They define in terms of another variable their functions of 00:21:38.200 --> 00:21:44.791 another variable. So if I use function of a function DY by DT 00:21:44.791 --> 00:21:47.326 times DT by The X. 00:21:48.000 --> 00:21:50.665 Then I can calculate the 00:21:50.665 --> 00:21:57.910 gradient. Divide by DT and let's remember that DT by DX 00:21:57.910 --> 00:22:05.146 is the same as one over the X by DT. In other 00:22:05.146 --> 00:22:11.779 words, D why by DX divided by the X by DT. 00:22:13.470 --> 00:22:18.470 Divide by DT is the derivative of this. 00:22:19.990 --> 00:22:24.742 The xpi DT is going to be the derivative of that. 00:22:25.760 --> 00:22:29.546 Divide by DT is just to 00:22:29.546 --> 00:22:36.863 A. Divided by the derivative of X and the derivative 00:22:36.863 --> 00:22:40.248 of that is 2 AT. 00:22:40.260 --> 00:22:43.949 Which gives us a gradient of one 00:22:43.949 --> 00:22:49.639 over T. So now we've got the gradient curve at that 00:22:49.639 --> 00:22:54.691 point, so we've got the gradient of the straight line, and we've 00:22:54.691 --> 00:23:01.427 got the point, so Y minus Y one over X Minus X one is equal to 00:23:01.427 --> 00:23:05.637 the gradient, so that's a standard formula for finding the 00:23:05.637 --> 00:23:10.689 equation of a straight line given a point on the line X1Y 00:23:10.689 --> 00:23:15.741 one and it's gradient. So let's substitute in the things that we 00:23:15.741 --> 00:23:18.514 know. This is going to be our X 00:23:18.514 --> 00:23:23.537 one. This is going to be our why one because that's the point on 00:23:23.537 --> 00:23:28.143 the curve. This is going to be our value of M our gradient, so 00:23:28.143 --> 00:23:31.762 let's just write down that information again. So X one we 00:23:31.762 --> 00:23:36.039 said was going to be the point on the curve X equals 80 00:23:36.039 --> 00:23:43.380 squared. Our Y one was going to be to AT and our gradient 00:23:43.380 --> 00:23:50.870 M, which was the why by DX was going to be one over T. 00:23:50.870 --> 00:23:56.220 The standard equation of a straight line given a point 00:23:56.220 --> 00:24:02.105 on the line X1Y one and it's gradient M. Now let's 00:24:02.105 --> 00:24:07.455 substitute that information in so we have Y minus 280. 00:24:08.780 --> 00:24:15.836 All over X minus 8 Y squared is equal to one over 00:24:15.836 --> 00:24:21.894 T. Now we need to multiply everything by T in order to get 00:24:21.894 --> 00:24:26.206 tea out of the denominator here and multiply everything by X 00:24:26.206 --> 00:24:32.086 minus 8 Y squared in order to get the X minus 8 Y squared out 00:24:32.086 --> 00:24:36.398 of the denominator. So and multiply everything by T so we 00:24:36.398 --> 00:24:41.102 have T times Y minus 280 is equal to and we multiply 00:24:41.102 --> 00:24:45.806 everything by X minus 8 Y squared and on this side it 00:24:45.806 --> 00:24:47.374 means we're multiplying by. 00:24:47.400 --> 00:24:54.636 One, so that's just X minus 8 Y squared. Multiply out this 00:24:54.636 --> 00:25:01.872 bracket. Ty minus 280 squared is equal to X minus AT squared. 00:25:02.680 --> 00:25:09.042 Now. Here we've got 80 squared term in 80 squared minus 280 00:25:09.042 --> 00:25:13.904 squared. And here we've got another term in 80 squared which 00:25:13.904 --> 00:25:20.092 is just minus 80 squared. So if I add 280 squared to each side, 00:25:20.092 --> 00:25:25.396 I'll just have 80 squared on this side and this will just 00:25:25.396 --> 00:25:31.142 leave me with T. Why? So that's Ty equals X plus 80 squared. 00:25:31.740 --> 00:25:37.570 And that's the equation of the tangent at the point P. 00:25:38.340 --> 00:25:43.081 Now there is a reason for working out this tangent. It's 00:25:43.081 --> 00:25:47.822 not just an exercise in using some calculus and using some 00:25:47.822 --> 00:25:52.563 coordinate geometry. I've done it for a purpose. I want to 00:25:52.563 --> 00:25:55.580 explore the reflective property of a parabola. 00:25:56.210 --> 00:26:00.656 So first of all, let's think what that actually 00:26:00.656 --> 00:26:01.644 is reflection. 00:26:04.520 --> 00:26:11.030 If we have a flat plane surface, then if a beam of light comes 00:26:11.030 --> 00:26:16.145 into that surface at an angle theater, the law of reflection 00:26:16.145 --> 00:26:21.725 tells us that it's reflected out again at exactly the same angle 00:26:21.725 --> 00:26:26.840 to the surface. So it comes in it's reflected out again. 00:26:27.790 --> 00:26:33.775 What if we had not a plain flat surface? 00:26:34.290 --> 00:26:36.018 But let's say a. 00:26:36.710 --> 00:26:42.924 Curved surface well what the law says is, I'm sorry, but a plane 00:26:42.924 --> 00:26:48.660 for a curved surface is exactly the same as a plain flat 00:26:48.660 --> 00:26:53.918 surface, except we take the tangent at the point where it's 00:26:53.918 --> 00:27:00.132 coming in. So the Ray of light is reflected out at the same 00:27:00.132 --> 00:27:05.762 angle. To the tangent that it made to the tangent when it came 00:27:05.762 --> 00:27:09.422 in. So that's the reflective 00:27:09.422 --> 00:27:16.060 law. How does this affect a parabola? Say we had a 00:27:16.060 --> 00:27:21.791 parabolic mirror mirror that was in the shape of our parabola. 00:27:22.590 --> 00:27:29.701 As our parabola, what happens to a Ray of light that's a comes 00:27:29.701 --> 00:27:36.812 in parallel to the X axis, and it strikes the mirror when it 00:27:36.812 --> 00:27:40.094 comes in, it must be reflected. 00:27:41.060 --> 00:27:46.536 Question is. Where is it reflected? What direction does 00:27:46.536 --> 00:27:50.758 that going? The law of reflection tells us. 00:27:52.070 --> 00:27:55.832 That if that's the angle theater that's made there, 00:27:55.832 --> 00:28:00.012 then that must be the angle theater that's made there. 00:28:01.080 --> 00:28:04.069 So what we need to discover is. 00:28:05.430 --> 00:28:10.260 If this is the case, where does that go? Where is that Ray of 00:28:10.260 --> 00:28:15.090 light directed? And as you can see, this is all to do with the 00:28:15.090 --> 00:28:18.885 tangent to the curve, so it's quite important that we know 00:28:18.885 --> 00:28:21.645 what the equation of that particular tangent is. 00:28:22.170 --> 00:28:23.900 OK. 00:28:24.970 --> 00:28:28.274 Now we know what it is we're trying to do. Let's set up a 00:28:28.274 --> 00:28:32.258 diagram. And on this diagram, let's put some. 00:28:33.670 --> 00:28:39.196 Coordinates of important things, so there's the direct tricks, 00:28:39.196 --> 00:28:42.880 and there's our focus. 2 very 00:28:42.880 --> 00:28:45.556 important points. Here's our 00:28:45.556 --> 00:28:47.519 parabola. Whoops, a Daisy. 00:28:48.690 --> 00:28:51.870 Better curve, here's our point, P. 00:28:54.300 --> 00:29:00.240 There and here's our tangent to the point P. 00:29:00.410 --> 00:29:05.811 I'll call that the point T there an array of light. 00:29:06.670 --> 00:29:13.432 Going to come in like so I'm going to do. I'm going to join 00:29:13.432 --> 00:29:15.364 that to the focus. 00:29:16.060 --> 00:29:20.500 So here I've got this line here, which is parallel. 00:29:23.600 --> 00:29:29.390 To the X axis I'm going to call this end, so I've got a line 00:29:29.390 --> 00:29:34.794 here, PN. If I extend PN back in a straight line, it's going to 00:29:34.794 --> 00:29:40.198 meet the direct tricks there. At the point M and there will be a 00:29:40.198 --> 00:29:44.758 right angle. But remember what the 00:29:44.758 --> 00:29:49.300 point Pierce .80 squared to AT. 00:29:50.640 --> 00:29:52.980 OK. 00:29:54.710 --> 00:30:01.430 One of the things that we need to find is this point T down 00:30:01.430 --> 00:30:06.230 here, let's remember the equation of our tangent. It's Ty 00:30:06.230 --> 00:30:11.990 is equal to X plus 80 squared. That's the equation of the 00:30:11.990 --> 00:30:15.456 Tangent. PT. So 00:30:15.456 --> 00:30:21.120 at T. Y is equal to 0. 00:30:21.800 --> 00:30:28.730 That implies that X is equal to minus AT. 00:30:29.610 --> 00:30:30.390 Square. 00:30:33.020 --> 00:30:40.588 That enables us to find this length here 00:30:40.588 --> 00:30:48.030 TF. Must be that distance up to the origin. Oh, which 00:30:48.030 --> 00:30:55.169 is a distance 80 squared plus that. So that's a plus 00:30:55.169 --> 00:30:56.467 AT squared. 00:30:58.790 --> 00:31:06.374 We know that PF is equal to P. 00:31:06.910 --> 00:31:08.940 N. 00:31:10.530 --> 00:31:15.894 Because that comes from the definition of the parabola. 00:31:18.160 --> 00:31:25.870 But we also know what PM actually is. PM is 00:31:25.870 --> 00:31:33.580 this distance A plus? This distance here, which is 80 00:31:33.580 --> 00:31:39.804 squared. So if we look at the argument that we've got here, 00:31:39.804 --> 00:31:46.720 that TF is A plus 80 squared PM is A plus. 8880 squared and 00:31:46.720 --> 00:31:53.636 PM is equal to PF. And this means that TF and PF have to 00:31:53.636 --> 00:31:55.612 be the same length. 00:31:56.650 --> 00:32:00.226 In other words, this triangle here. 00:32:01.240 --> 00:32:03.168 Is an isosceles triangle. 00:32:06.070 --> 00:32:08.610 Triangle. 00:32:09.850 --> 00:32:12.480 FTP 00:32:12.480 --> 00:32:15.110 is. 00:32:15.470 --> 00:32:22.370 Isosceles OK, it's isosceles. What does 00:32:22.370 --> 00:32:28.514 that tell us? Well, if it's isosceles, it tells us that it's 00:32:28.514 --> 00:32:34.146 two base angles are equal, so that's one equal side, and 00:32:34.146 --> 00:32:40.802 that's one equal side, so that angle has to be equal to that 00:32:40.802 --> 00:32:46.434 angle there. Therefore, the angle FTP is equal to the angle 00:32:46.434 --> 00:32:50.018 TPF. Let me just mark those that 00:32:50.018 --> 00:32:52.870 angle there. Is Alpha. 00:32:53.880 --> 00:32:57.186 And that angle there is Alpha. 00:32:58.720 --> 00:33:01.080 But wait a minute. 00:33:02.240 --> 00:33:04.862 This line and this line are 00:33:04.862 --> 00:33:11.994 parallel. So therefore, this angle NPF is equal 00:33:11.994 --> 00:33:15.362 to this angle PF. 00:33:16.090 --> 00:33:21.850 T. Those two angles are both equal to. 00:33:22.660 --> 00:33:23.720 Beta. 00:33:25.550 --> 00:33:30.269 Be cause this line is parallel to that one, they make a zed 00:33:30.269 --> 00:33:35.876 angle. Now the things of this triangle add up to 180 and 00:33:35.876 --> 00:33:40.868 here is a straight line, the angles of which must add up 00:33:40.868 --> 00:33:45.028 to 180, and so this angle here is also Alpha. 00:33:46.410 --> 00:33:48.450 Hang on a minute. 00:33:49.390 --> 00:33:53.702 What we're saying is that the angle that this line makes 00:33:53.702 --> 00:33:58.014 with the tangent is equal to the angle that this line 00:33:58.014 --> 00:33:59.582 makes with the Tangent. 00:34:01.430 --> 00:34:06.904 In other words, if this was a Ray of light coming in here, it 00:34:06.904 --> 00:34:11.205 would be reflected according to the law of reflection, and it 00:34:11.205 --> 00:34:13.160 would pass through the focus. 00:34:14.600 --> 00:34:21.880 So any Ray of light that comes in parallel to the X axis? 00:34:22.710 --> 00:34:26.526 Each Ray of light is reflected to the focus. 00:34:27.240 --> 00:34:32.412 Now you can see how your dish aerial works. Your dish aerial 00:34:32.412 --> 00:34:37.153 is formed by spinning a parabola, so it makes a surface 00:34:37.153 --> 00:34:42.325 a dish. The signal comes in and strikes the dish and is 00:34:42.325 --> 00:34:47.066 reflected to the receptor. That little lump that stands up in 00:34:47.066 --> 00:34:52.669 front of the disk and the whole of the signal is gathered there. 00:34:53.930 --> 00:34:59.292 How else can we might use of this? What we can make use of? 00:34:59.292 --> 00:35:04.271 Its in Searchlight 'cause we can turn it around the other way. If 00:35:04.271 --> 00:35:09.633 we put a bulb there at the focus and it emits light then the 00:35:09.633 --> 00:35:14.229 light that he meets will travel to the parabolic mirror an will 00:35:14.229 --> 00:35:18.442 be reflected outwards in a beam. A concentrated beam, not one 00:35:18.442 --> 00:35:22.655 that spreads but one that is concentrated and is parallel to 00:35:22.655 --> 00:35:24.570 the X axis to the. 00:35:24.620 --> 00:35:25.940 Axis of the mirror. 00:35:26.630 --> 00:35:31.869 So we see that are a property of a curve discovered and known 00:35:31.869 --> 00:35:35.899 about by the ancient Greeks has some very, very modern 00:35:35.899 --> 00:35:40.332 applications, and indeed our SOC would not be the same without 00:35:40.332 --> 00:35:44.362 the kinds of properties that we're talking about now that 00:35:44.362 --> 00:35:46.377 exist in these conic sections.