0:00:02.440,0:00:07.913
This particular video is going[br]to be about of a set of curves

0:00:07.913,0:00:09.597
known as conic sections.

0:00:10.290,0:00:12.540
Here's a cone.

0:00:13.080,0:00:19.151
And conic sections are formed by[br]taking cuts in into this cone in

0:00:19.151,0:00:23.830
various ways. Now, the curves[br]that are formed by taking these

0:00:23.830,0:00:28.030
particular cuts or sections are[br]very old. They were known to the

0:00:28.030,0:00:29.780
ancient Greeks from about 3:50

0:00:29.780,0:00:35.244
BC onwards. But not only they[br]very old, but they also a very

0:00:35.244,0:00:39.709
modern. The dish aerial that's[br]outside your house that brings

0:00:39.709,0:00:44.010
in Sky Television is based upon[br]the reflective property of one

0:00:44.010,0:00:48.702
of these conic sections that[br]we're going to be having a look

0:00:48.702,0:00:52.221
at Georgia Bank Telescope, a[br]radar telescope that pierces

0:00:52.221,0:00:56.131
deep into the universe, picking[br]up signals, recording them for

0:00:56.131,0:01:00.432
us again, is based upon a[br]reflective property of one of

0:01:00.432,0:01:03.951
these conic sections. So[br]although they are ancient, even

0:01:03.951,0:01:06.297
though they are, they have very,

0:01:06.297,0:01:08.418
very. Important modern uses.

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So let's have a look at our[br]coal. Let's begin by describing

0:01:13.246,0:01:17.074
some features about the cone[br]itself, which I'll need in order

0:01:17.074,0:01:21.250
to describe the sections and the[br]curves that we're going to see.

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First of all, let's think about[br]the axis of the cone. Now,

0:01:25.426,0:01:29.254
that's the line that runs from[br]the point down through the

0:01:29.254,0:01:32.734
center of the basin out there.[br]So that's the axis.

0:01:33.360,0:01:37.824
The generator of the comb. Well,[br]that's any line that runs down

0:01:37.824,0:01:42.660
the side of the cone like that[br]and the generator. Think of it

0:01:42.660,0:01:47.124
as being a pendulum, a piece of[br]string tight there. Its wings

0:01:47.124,0:01:51.588
round here and it Maps out this[br]curved surface of the cone.

0:01:52.480,0:01:57.745
So let's begin by looking at the[br]cuts or sections that we can

0:01:57.745,0:02:02.606
make. One of the simplest[br]sections that we can make is to

0:02:02.606,0:02:06.878
cut across and perpendicular to[br]the axis of the cone. And what

0:02:06.878,0:02:10.438
do we get? We get that well[br]known curve circle.

0:02:11.550,0:02:17.100
OK, let's not now cut[br]perpendicular. Let's cut at an

0:02:17.100,0:02:20.960
angle. So we're going to cut at[br]an angle to the.

0:02:21.780,0:02:23.468
Access of the cone.

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We're going to keep the cut[br]within the physical dimensions

0:02:27.640,0:02:32.385
of the cone. And what do we get?[br]We get a curved surface.

0:02:33.120,0:02:37.509
A closed curve again, this time[br]it's an ellipse, kind of

0:02:37.509,0:02:41.898
flattened circle if you like,[br]but its name is an ellipse.

0:02:42.800,0:02:44.210
Let's put that back on.

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And now let's take a cut[br]which is parallel till the

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generator parallel to the edge

0:02:54.006,0:02:59.386
of the. Curved surface, so[br]there's our edge.

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Of the curve surface, and this[br]is the cut that we're going to

0:03:05.639,0:03:09.669
take parallel to that edge. So[br]we make the cut.

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And there we've got an open[br]curve, this time, not a closed

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one. It's an open curve and it's[br]called the parabola.

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Putting that back together, what[br]I want you to imagine now is a

0:03:23.957,0:03:29.683
double Cole. Now what do I mean[br]by a double cone? I mean another

0:03:29.683,0:03:32.546
cone sitting on top of this one.

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Point to point. So imagine we've[br]got this cone here. We've got

0:03:38.770,0:03:40.550
another cone up here.

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And again, I'm going to take a[br]cut, which this time is going to

0:03:46.318,0:03:50.782
be parallel to the axis of the[br]coal coming straight down there,

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and you can see the cut there[br]now. Once I make it there, I've

0:03:55.990,0:04:00.454
made the cut will actually be[br]making the cut through the cone

0:04:00.454,0:04:04.918
up here, and so they'll actually[br]be 2 pieces to the curve.

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One from the upper cone and one[br]from the lower cone, and that's

0:04:11.646,0:04:15.102
a hyperbola. It's got two[br]branches to it.

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So those are our three conic[br]sections, sorry, miscounted.

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Those are our four conic[br]sections, the circle.

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The ellipse.

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The parabola.[br]On the hyperbola with

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its two branches.

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Now the conic sections were[br]first discovered by monarchists

0:04:42.384,0:04:47.532
in about 350 BC. Minarchist was[br]an ancient Greek. He in fact had

0:04:47.532,0:04:52.284
three different cones, but he[br]took the same cut in each cone

0:04:52.284,0:04:54.660
in order to generate three of

0:04:54.660,0:04:58.194
them. Here with disregarding the[br]circle, 'cause, That's obviously

0:04:58.194,0:04:59.809
the simplest cut to make.

0:05:00.540,0:05:05.810
Later Apollonius who was around[br]about 200 BC, generated this

0:05:05.810,0:05:11.607
idea of making the cuts all in[br]exactly the same cone.

0:05:12.340,0:05:17.715
So. We've seen what these[br]curves look like, and we've

0:05:17.715,0:05:23.110
seen how they are generated[br]now. What we want to do is to

0:05:23.110,0:05:24.770
explore their mathematical[br]equations.

0:05:25.790,0:05:28.913
If we don't explore these[br]mathematical equations, then we

0:05:28.913,0:05:33.424
won't be able to deal with these[br]curves and find out what their

0:05:33.424,0:05:37.720
properties are. So let's have a[br]look at a definition of these

0:05:37.720,0:05:40.474
curves that will actually help[br]us to calculate them.

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What I want first of all is a[br]straight line. Straight line has

0:05:45.300,0:05:50.020
a name. It's called[br]the direct tricks.

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And I want a fixed point. I'm[br]going to label that fixed point

0:05:58.366,0:06:01.176
F and it's F4 focus.

0:06:03.200,0:06:05.780
Take any point.

0:06:06.490,0:06:10.423
In the plane, let's call that[br]the point P.

0:06:11.600,0:06:17.723
And then we've got P must be a[br]distance away from this fix

0:06:17.723,0:06:23.375
straight line. Let's call that[br]point M and that's of course a

0:06:23.375,0:06:29.027
right angle. And it's also a[br]distance away from F. The focus.

0:06:29.600,0:06:35.324
And so the definition that we're[br]going to use is that PF.

0:06:35.870,0:06:39.176
The distance of the point from

0:06:39.176,0:06:42.909
the focus. Is equal to.

0:06:43.460,0:06:48.652
A constant multiple of the[br]distance of the point from the

0:06:48.652,0:06:54.316
straight line and that constant[br]multiple. We use the letter E to

0:06:54.316,0:07:00.390
denote. And so PF is equal[br]to E times by PM.

0:07:01.370,0:07:06.698
Now what happens? Well, this[br]point P follows what we call a

0:07:06.698,0:07:12.026
locus. Let's just write down[br]that word locus. What do we mean

0:07:12.026,0:07:18.242
by a locus? Well, it's simply a[br]path. The Point P follows a path

0:07:18.242,0:07:20.018
governed by this rule.

0:07:21.100,0:07:28.540
This letter EE is used[br]to stand for what we

0:07:28.540,0:07:30.772
call the eccentricity.

0:07:31.600,0:07:37.536
Of the curve and we[br]get our different.

0:07:38.080,0:07:43.976
Conic sections for different[br]values of E. So if E is

0:07:43.976,0:07:50.944
between North and won, the curve[br]that we get is the closed curve.

0:07:51.640,0:07:53.060
The ellipse.

0:07:54.100,0:07:58.940
If E is equal to 1, the curve[br]that we get.

0:07:59.740,0:08:01.210
Is the parabola.

0:08:01.830,0:08:04.756
The open curve but with a single

0:08:04.756,0:08:11.246
branch. And if he is greater[br]than one, the curve that we get

0:08:11.246,0:08:16.622
is the hyperbola. Remember that[br]was the open curve with the two

0:08:16.622,0:08:21.258
branches. Now what happens[br]in the rest of the video?

0:08:21.258,0:08:24.154
We're going to deal with[br]this particular curve.

0:08:26.060,0:08:31.182
In some detail we look at some[br]of its properties and will just

0:08:31.182,0:08:35.122
relat say what the equivalent[br]properties are in these two

0:08:35.122,0:08:40.244
curves. But this is the one that[br]we're going to deal with in

0:08:40.244,0:08:44.578
detail, and the sorts of[br]mathematics that we use and the

0:08:44.578,0:08:48.912
ways in which we work if[br]replicated will enable the same

0:08:48.912,0:08:53.246
sorts of properties to be proved[br]and established in the ellipse

0:08:53.246,0:08:55.610
and the hyperbola. So let's go

0:08:55.610,0:08:58.218
on. Have a look at the parabola.

0:08:59.520,0:09:02.288
So. Let's draw a picture.

0:09:04.820,0:09:09.960
So I'm going to set up some[br]coordinate axes X&Y.

0:09:10.650,0:09:14.410
I'm going to take my straight[br]line my direct tricks.

0:09:15.300,0:09:18.028
There, and I'm going to[br]take my focus.

0:09:19.050,0:09:25.291
There. Now I need some measure,[br]some scale, some size, so I'm

0:09:25.291,0:09:30.625
going to say I'm going to take[br]the focus at the point a nought.

0:09:31.910,0:09:37.520
Now remember that[br]4 hour parabola.

0:09:38.540,0:09:45.570
PF is equal to PM[br]be'cause E the eccentricity was

0:09:45.570,0:09:47.679
equal to 1.

0:09:48.310,0:09:52.899
So the distance of our point P[br]has to be the same from.

0:09:53.670,0:09:59.180
This line. As it is from this[br]point and of course, there's an

0:09:59.180,0:10:02.240
obvious point, namely the[br]origin, that perhaps we'd like

0:10:02.240,0:10:06.320
to be on this curve. So that[br]means that the direct tricks

0:10:06.320,0:10:09.040
here is going to be at X equals

0:10:09.040,0:10:12.108
minus a. Let's put our.

0:10:12.890,0:10:16.060
Point P. Let's say there.

0:10:17.380,0:10:18.720
There is.

0:10:19.970,0:10:27.076
PM That's our[br]point. N there is PF

0:10:27.076,0:10:31.684
will call this the point[br]XY.

0:10:33.280,0:10:38.128
So what's the locus? What path?[br]What's the equation of that path

0:10:38.128,0:10:42.976
that P is going to follow as it[br]moves according to this

0:10:42.976,0:10:47.824
definition, one of the things we[br]better do is we better write

0:10:47.824,0:10:49.844
down what are these lens?

0:10:50.600,0:10:57.838
PM equals well from M to the[br]Y axis is a distance A and

0:10:57.838,0:11:04.559
there's a further distance of X[br]to go before we get to pee.

0:11:04.559,0:11:07.661
So PM is A plus X.

0:11:08.340,0:11:11.790
What about PF?

0:11:12.900,0:11:18.204
Once we look at PF, it's the[br]hypotenuse of a right angle

0:11:18.204,0:11:22.624
triangle which we can form by[br]dropping a perpendicular down

0:11:22.624,0:11:25.718
there so we can see that PF.

0:11:27.060,0:11:31.656
Will be when we square it.[br]PF squared will be equal to

0:11:31.656,0:11:34.720
that squared, which is just[br]the height, why?

0:11:35.860,0:11:39.760
Plus that squared.

0:11:40.390,0:11:47.615
Will up to there is X and up to[br]there is a so that little bit in

0:11:47.615,0:11:49.740
there is a minus X.

0:11:50.340,0:11:51.360
Squared

0:11:54.620,0:11:58.080
So this is PF squared.

0:11:58.670,0:12:03.734
Let's just check that again, PF[br]is the hypotenuse of a right

0:12:03.734,0:12:07.532
angle triangle, so using[br]Pythagoras PF squared is that

0:12:07.532,0:12:11.752
squared plus that squared.[br]That's the height YP is above

0:12:11.752,0:12:13.018
the X axis.

0:12:13.640,0:12:19.880
Up to there is X up to. There is[br]a, so the distance between there

0:12:19.880,0:12:22.376
and there is a minus X.

0:12:22.950,0:12:28.943
So now I need to equate these[br]two expressions and first of all

0:12:28.943,0:12:35.397
that means I've got to square PM[br]because if PF is equal to PMPF

0:12:35.397,0:12:41.964
squared. Must be equal to PM[br]squared and so we can substitute

0:12:41.964,0:12:49.580
this in so instead of PF we[br]have Y squared A minus, X all

0:12:49.580,0:12:56.108
squared and instead of PM[br]squared we have a plus X all

0:12:56.108,0:13:02.732
squared. Now we need to[br]multiply out this bracket Y

0:13:02.732,0:13:08.414
squared plus. Now let's just do[br]the multiplication over here.

0:13:08.920,0:13:15.318
A minor sex, all squared is a[br]minus X times by A minus X.

0:13:15.890,0:13:20.587
So we've a Times by a that gives[br]us a squared.

0:13:21.330,0:13:27.210
With a Times by minus X and[br]minus X times by a, which gives

0:13:27.210,0:13:33.510
us minus two AX and then we have[br]minus X times Y minus X which

0:13:33.510,0:13:38.550
gives us plus X squared equals.[br]Let's have a look at this

0:13:38.550,0:13:43.590
bracket. It's a plus X all[br]squared, so it's going to be

0:13:43.590,0:13:48.630
exactly the same as this one,[br]except with plus signs in. So

0:13:48.630,0:13:51.570
we're going to get a squared +2.

0:13:51.600,0:13:55.100
X plus X squared.

0:13:56.360,0:14:00.980
Now let's have a look at this.[br]We've A plus a squared here and

0:14:00.980,0:14:05.930
a plus a squared there so we can[br]take an A squared away from each

0:14:05.930,0:14:10.220
side. We have a plus X squared[br]here under plus X squared here

0:14:10.220,0:14:12.860
so we can take an X squared away

0:14:12.860,0:14:19.276
from each side. We've minus two[br]X here and plus 2X there, so it

0:14:19.276,0:14:24.508
makes sense to get those axis[br]together by adding 2X to this

0:14:24.508,0:14:27.560
side and adding it to that side.

0:14:28.070,0:14:31.710
So if we do all that[br]we've got Y squared.

0:14:33.140,0:14:37.901
The two A squared will[br]disappear, subtracting a square

0:14:37.901,0:14:42.662
from each side. The two X[br]squared will disappear.

0:14:42.662,0:14:49.010
Subtracting X squared from each[br]side and adding the two X2 each

0:14:49.010,0:14:54.829
side, we get 4A X that is[br]the standard Cartesian equation

0:14:54.829,0:14:56.416
for a parabola.

0:14:57.500,0:14:59.040
Now.

0:15:00.400,0:15:04.274
This is a standard equation and[br]it is the equation with which we

0:15:04.274,0:15:08.826
want to work. But sometimes[br]when we're doing these

0:15:08.826,0:15:13.560
curves, it's helpful to have[br]the equation described in

0:15:13.560,0:15:15.664
terms of 1/3 variable.

0:15:17.110,0:15:22.330
Variable that's often[br]called a parameter.

0:15:23.480,0:15:28.100
And the parameter we're going to[br]have in this case is the

0:15:28.100,0:15:29.630
parameter. T.

0:15:30.840,0:15:37.880
So we've got an equation[br]Y squared equals 4A X.

0:15:38.680,0:15:43.650
And what we're looking for is a[br]way of expressing X in terms of

0:15:43.650,0:15:47.555
tea and a way of expressing Y in[br]terms of T.

0:15:48.590,0:15:54.372
When we look at this, this says[br]Y squared equals 4A X. Now Y

0:15:54.372,0:15:58.915
squared is a complete square.[br]It's a whole square an exact

0:15:58.915,0:16:03.871
square. I can take it square[br]root and I'll just get plus

0:16:03.871,0:16:09.653
online as why so? Can I do it[br]over here? Can I choose an

0:16:09.653,0:16:13.783
expression for X that will give[br]me a complete square?

0:16:14.480,0:16:18.560
On this side of the equation,[br]well, four is already a complete

0:16:18.560,0:16:23.320
square. If I want to make a a[br]complete square and have to have

0:16:23.320,0:16:27.740
a squared. So I need something[br]in X that's gotten a attached to

0:16:27.740,0:16:31.480
it. So 8 times by a would give[br]me a squared.

0:16:32.070,0:16:36.295
And then I want to be able to[br]take this exact square root.

0:16:36.295,0:16:39.545
Will the only thing if I'm[br]going to introduce this

0:16:39.545,0:16:42.795
variable T that suggests[br]itself and he sensible is T

0:16:42.795,0:16:44.420
squared? So if we put.

0:16:45.990,0:16:52.510
X equals AT squared,[br]then Y squared is

0:16:52.510,0:16:59.030
equal to four a[br]Times 80 squared, which

0:16:59.030,0:17:05.550
is 4A squared T[br]squared. And so why

0:17:05.550,0:17:08.810
is equal to two

0:17:08.810,0:17:14.350
AT? And So what I've got[br]here is what's called

0:17:14.350,0:17:15.670
the parametric equation.

0:17:16.830,0:17:20.220
For the parabola.

0:17:22.100,0:17:24.540
Now that's our parametric

0:17:24.540,0:17:31.269
equation. And this is our[br]Cartesian Equation. One thing we

0:17:31.269,0:17:34.274
haven't done yet is sketched.

0:17:34.910,0:17:40.022
The curve itself, so just want[br]to do that.

0:17:41.890,0:17:42.919
So there's a.

0:17:43.600,0:17:45.580
X&Y axis.

0:17:46.830,0:17:50.386
There's our direct tricks[br]through minus a.

0:17:51.130,0:17:54.718
Here's our focus at a nought.

0:17:55.930,0:18:00.597
We know the curve is going to go[br]through there and it's equation

0:18:00.597,0:18:05.623
is Y squared equals 4A X. Well,[br]if we take positive values of X,

0:18:05.623,0:18:10.649
we can see we're going to get[br]that, but a positive value of X

0:18:10.649,0:18:15.675
gives us the value of Y, which[br]when we take the square root is

0:18:15.675,0:18:20.342
plus or minus. So we can see[br]we're going to have that and.

0:18:20.870,0:18:25.787
Symmetry in the X axis. Notice[br]we can't have any negative

0:18:25.787,0:18:31.598
values of X because 4A A is a[br]positive number times a negative

0:18:31.598,0:18:36.962
value of X would give us a[br]negative value for Y squared.

0:18:37.810,0:18:41.960
What about the parametric[br]equation X equals 80 squared Y

0:18:41.960,0:18:47.355
equals 280? What does this do in[br]terms of this picture? Well, if

0:18:47.355,0:18:53.165
we look at it, we can see X[br]equals 80 squared. A is a

0:18:53.165,0:18:57.315
positive number, T squared is[br]positive, so again we've only

0:18:57.315,0:19:02.295
got values of X which are[br]greater than or equal to 0.

0:19:03.470,0:19:09.630
However, values of why can range[br]from minus Infinity to plus

0:19:09.630,0:19:15.790
Infinity? And if you like tea[br]down here is minus Infinity.

0:19:16.930,0:19:22.910
Going round to T equals 0 here[br]at the origin coming round. If

0:19:22.910,0:19:26.130
you like to T is plus Infinity

0:19:26.130,0:19:33.015
there. So T is a parameter and[br]in a sense it counts is around

0:19:33.015,0:19:37.800
the curve as T increases from[br]minus Infinity to plus Infinity,

0:19:37.800,0:19:39.975
we move around the curve.

0:19:40.490,0:19:47.060
OK. What[br]about this curve? Well, one of

0:19:47.060,0:19:53.300
the things that we want to be[br]able to establish is what's the

0:19:53.300,0:19:58.580
equation of the tangent to the[br]curve at a particular point.

0:20:00.260,0:20:05.220
So again, let's draw our[br]picture of the curve, putting

0:20:05.220,0:20:10.180
in the direct tricks straight[br]line, putting in the focus

0:20:10.180,0:20:11.668
and putting in.

0:20:12.690,0:20:13.670
The curve.

0:20:15.200,0:20:20.680
Let's take any point P[br]on the curve.

0:20:21.380,0:20:27.373
And the question we are asking[br]is what is the equation of the

0:20:27.373,0:20:29.217
tangent to the curve?

0:20:30.500,0:20:37.234
What's the equation of that lie?[br]Well, I've got P is any point on

0:20:37.234,0:20:43.006
the curve. So instead of calling[br]XY, I'm going to use the

0:20:43.006,0:20:49.276
parametric equation. X is 80[br]squared, Y is 280.

0:20:50.150,0:20:56.855
So X is 80 squared,[br]Y is 2 AT.

0:20:57.750,0:21:01.996
The question we're asking what[br]is the equation of the tangent?

0:21:01.996,0:21:06.628
Well, it's a straight line to[br]find the equation of a straight

0:21:06.628,0:21:12.032
line, we need a point on that[br]line or we've got it 80 squared

0:21:12.032,0:21:17.436
280 and we need the gradient of[br]the line and the gradient of the

0:21:17.436,0:21:22.068
line. Of course must be the[br]gradient of the curve at that

0:21:22.068,0:21:25.156
point P, so we need DY by X.

0:21:26.680,0:21:32.359
But both X&Y are defined in[br]terms of T.

0:21:33.130,0:21:38.200
They define in terms of another[br]variable their functions of

0:21:38.200,0:21:44.791
another variable. So if I use[br]function of a function DY by DT

0:21:44.791,0:21:47.326
times DT by The X.

0:21:48.000,0:21:50.665
Then I can calculate the

0:21:50.665,0:21:57.910
gradient. Divide by DT and[br]let's remember that DT by DX

0:21:57.910,0:22:05.146
is the same as one over[br]the X by DT. In other

0:22:05.146,0:22:11.779
words, D why by DX divided[br]by the X by DT.

0:22:13.470,0:22:18.470
Divide by DT is the[br]derivative of this.

0:22:19.990,0:22:24.742
The xpi DT is going to be the[br]derivative of that.

0:22:25.760,0:22:29.546
Divide by DT is just to

0:22:29.546,0:22:36.863
A. Divided by the derivative[br]of X and the derivative

0:22:36.863,0:22:40.248
of that is 2 AT.

0:22:40.260,0:22:43.949
Which gives us a gradient of one

0:22:43.949,0:22:49.639
over T. So now we've got[br]the gradient curve at that

0:22:49.639,0:22:54.691
point, so we've got the gradient[br]of the straight line, and we've

0:22:54.691,0:23:01.427
got the point, so Y minus Y one[br]over X Minus X one is equal to

0:23:01.427,0:23:05.637
the gradient, so that's a[br]standard formula for finding the

0:23:05.637,0:23:10.689
equation of a straight line[br]given a point on the line X1Y

0:23:10.689,0:23:15.741
one and it's gradient. So let's[br]substitute in the things that we

0:23:15.741,0:23:18.514
know. This is going to be our X

0:23:18.514,0:23:23.537
one. This is going to be our why[br]one because that's the point on

0:23:23.537,0:23:28.143
the curve. This is going to be[br]our value of M our gradient, so

0:23:28.143,0:23:31.762
let's just write down that[br]information again. So X one we

0:23:31.762,0:23:36.039
said was going to be the point[br]on the curve X equals 80

0:23:36.039,0:23:43.380
squared. Our Y one was going[br]to be to AT and our gradient

0:23:43.380,0:23:50.870
M, which was the why by DX[br]was going to be one over T.

0:23:50.870,0:23:56.220
The standard equation of a[br]straight line given a point

0:23:56.220,0:24:02.105
on the line X1Y one and it's[br]gradient M. Now let's

0:24:02.105,0:24:07.455
substitute that information[br]in so we have Y minus 280.

0:24:08.780,0:24:15.836
All over X minus 8 Y[br]squared is equal to one over

0:24:15.836,0:24:21.894
T. Now we need to multiply[br]everything by T in order to get

0:24:21.894,0:24:26.206
tea out of the denominator here[br]and multiply everything by X

0:24:26.206,0:24:32.086
minus 8 Y squared in order to[br]get the X minus 8 Y squared out

0:24:32.086,0:24:36.398
of the denominator. So and[br]multiply everything by T so we

0:24:36.398,0:24:41.102
have T times Y minus 280 is[br]equal to and we multiply

0:24:41.102,0:24:45.806
everything by X minus 8 Y[br]squared and on this side it

0:24:45.806,0:24:47.374
means we're multiplying by.

0:24:47.400,0:24:54.636
One, so that's just X minus[br]8 Y squared. Multiply out this

0:24:54.636,0:25:01.872
bracket. Ty minus 280 squared is[br]equal to X minus AT squared.

0:25:02.680,0:25:09.042
Now. Here we've got 80 squared[br]term in 80 squared minus 280

0:25:09.042,0:25:13.904
squared. And here we've got[br]another term in 80 squared which

0:25:13.904,0:25:20.092
is just minus 80 squared. So if[br]I add 280 squared to each side,

0:25:20.092,0:25:25.396
I'll just have 80 squared on[br]this side and this will just

0:25:25.396,0:25:31.142
leave me with T. Why? So that's[br]Ty equals X plus 80 squared.

0:25:31.740,0:25:37.570
And that's the equation of the[br]tangent at the point P.

0:25:38.340,0:25:43.081
Now there is a reason for[br]working out this tangent. It's

0:25:43.081,0:25:47.822
not just an exercise in using[br]some calculus and using some

0:25:47.822,0:25:52.563
coordinate geometry. I've done[br]it for a purpose. I want to

0:25:52.563,0:25:55.580
explore the reflective property[br]of a parabola.

0:25:56.210,0:26:00.656
So first of all, let's[br]think what that actually

0:26:00.656,0:26:01.644
is reflection.

0:26:04.520,0:26:11.030
If we have a flat plane surface,[br]then if a beam of light comes

0:26:11.030,0:26:16.145
into that surface at an angle[br]theater, the law of reflection

0:26:16.145,0:26:21.725
tells us that it's reflected out[br]again at exactly the same angle

0:26:21.725,0:26:26.840
to the surface. So it comes in[br]it's reflected out again.

0:26:27.790,0:26:33.775
What if we had not a[br]plain flat surface?

0:26:34.290,0:26:36.018
But let's say a.

0:26:36.710,0:26:42.924
Curved surface well what the law[br]says is, I'm sorry, but a plane

0:26:42.924,0:26:48.660
for a curved surface is exactly[br]the same as a plain flat

0:26:48.660,0:26:53.918
surface, except we take the[br]tangent at the point where it's

0:26:53.918,0:27:00.132
coming in. So the Ray of light[br]is reflected out at the same

0:27:00.132,0:27:05.762
angle. To the tangent that it[br]made to the tangent when it came

0:27:05.762,0:27:09.422
in. So that's the reflective

0:27:09.422,0:27:16.060
law. How does this affect[br]a parabola? Say we had a

0:27:16.060,0:27:21.791
parabolic mirror mirror that was[br]in the shape of our parabola.

0:27:22.590,0:27:29.701
As our parabola, what happens to[br]a Ray of light that's a comes

0:27:29.701,0:27:36.812
in parallel to the X axis, and[br]it strikes the mirror when it

0:27:36.812,0:27:40.094
comes in, it must be reflected.

0:27:41.060,0:27:46.536
Question is. Where is it[br]reflected? What direction does

0:27:46.536,0:27:50.758
that going? The law of[br]reflection tells us.

0:27:52.070,0:27:55.832
That if that's the angle[br]theater that's made there,

0:27:55.832,0:28:00.012
then that must be the angle[br]theater that's made there.

0:28:01.080,0:28:04.069
So what we need to discover is.

0:28:05.430,0:28:10.260
If this is the case, where does[br]that go? Where is that Ray of

0:28:10.260,0:28:15.090
light directed? And as you can[br]see, this is all to do with the

0:28:15.090,0:28:18.885
tangent to the curve, so it's[br]quite important that we know

0:28:18.885,0:28:21.645
what the equation of that[br]particular tangent is.

0:28:22.170,0:28:23.900
OK.

0:28:24.970,0:28:28.274
Now we know what it is we're[br]trying to do. Let's set up a

0:28:28.274,0:28:32.258
diagram. And on this diagram,[br]let's put some.

0:28:33.670,0:28:39.196
Coordinates of important things,[br]so there's the direct tricks,

0:28:39.196,0:28:42.880
and there's our focus. 2 very

0:28:42.880,0:28:45.556
important points. Here's our

0:28:45.556,0:28:47.519
parabola. Whoops, a Daisy.

0:28:48.690,0:28:51.870
Better curve, here's our[br]point, P.

0:28:54.300,0:29:00.240
There and here's our tangent to[br]the point P.

0:29:00.410,0:29:05.811
I'll call that the point T there[br]an array of light.

0:29:06.670,0:29:13.432
Going to come in like so I'm[br]going to do. I'm going to join

0:29:13.432,0:29:15.364
that to the focus.

0:29:16.060,0:29:20.500
So here I've got this line here,[br]which is parallel.

0:29:23.600,0:29:29.390
To the X axis I'm going to call[br]this end, so I've got a line

0:29:29.390,0:29:34.794
here, PN. If I extend PN back in[br]a straight line, it's going to

0:29:34.794,0:29:40.198
meet the direct tricks there. At[br]the point M and there will be a

0:29:40.198,0:29:44.758
right angle. But[br]remember what the

0:29:44.758,0:29:49.300
point Pierce .80[br]squared to AT.

0:29:50.640,0:29:52.980
OK.

0:29:54.710,0:30:01.430
One of the things that we need[br]to find is this point T down

0:30:01.430,0:30:06.230
here, let's remember the[br]equation of our tangent. It's Ty

0:30:06.230,0:30:11.990
is equal to X plus 80 squared.[br]That's the equation of the

0:30:11.990,0:30:15.456
Tangent. PT. So

0:30:15.456,0:30:21.120
at T. Y is[br]equal to 0.

0:30:21.800,0:30:28.730
That implies that X is[br]equal to minus AT.

0:30:29.610,0:30:30.390
Square.

0:30:33.020,0:30:40.588
That enables us to[br]find this length here

0:30:40.588,0:30:48.030
TF. Must be that distance[br]up to the origin. Oh, which

0:30:48.030,0:30:55.169
is a distance 80 squared[br]plus that. So that's a plus

0:30:55.169,0:30:56.467
AT squared.

0:30:58.790,0:31:06.374
We know that PF[br]is equal to P.

0:31:06.910,0:31:08.940
N.

0:31:10.530,0:31:15.894
Because that comes from the[br]definition of the parabola.

0:31:18.160,0:31:25.870
But we also know what[br]PM actually is. PM is

0:31:25.870,0:31:33.580
this distance A plus? This[br]distance here, which is 80

0:31:33.580,0:31:39.804
squared. So if we look at the[br]argument that we've got here,

0:31:39.804,0:31:46.720
that TF is A plus 80 squared[br]PM is A plus. 8880 squared and

0:31:46.720,0:31:53.636
PM is equal to PF. And this[br]means that TF and PF have to

0:31:53.636,0:31:55.612
be the same length.

0:31:56.650,0:32:00.226
In other words, this[br]triangle here.

0:32:01.240,0:32:03.168
Is an isosceles triangle.

0:32:06.070,0:32:08.610
Triangle.

0:32:09.850,0:32:12.480
FTP

0:32:12.480,0:32:15.110
is.

0:32:15.470,0:32:22.370
Isosceles[br]OK, it's isosceles. What does

0:32:22.370,0:32:28.514
that tell us? Well, if it's[br]isosceles, it tells us that it's

0:32:28.514,0:32:34.146
two base angles are equal, so[br]that's one equal side, and

0:32:34.146,0:32:40.802
that's one equal side, so that[br]angle has to be equal to that

0:32:40.802,0:32:46.434
angle there. Therefore, the[br]angle FTP is equal to the angle

0:32:46.434,0:32:50.018
TPF. Let me just mark those that

0:32:50.018,0:32:52.870
angle there. Is Alpha.

0:32:53.880,0:32:57.186
And that angle there is Alpha.

0:32:58.720,0:33:01.080
But wait a minute.

0:33:02.240,0:33:04.862
This line and this line are

0:33:04.862,0:33:11.994
parallel. So therefore, this[br]angle NPF is equal

0:33:11.994,0:33:15.362
to this angle PF.

0:33:16.090,0:33:21.850
T. Those two angles are[br]both equal to.

0:33:22.660,0:33:23.720
Beta.

0:33:25.550,0:33:30.269
Be cause this line is parallel[br]to that one, they make a zed

0:33:30.269,0:33:35.876
angle. Now the things of this[br]triangle add up to 180 and

0:33:35.876,0:33:40.868
here is a straight line, the[br]angles of which must add up

0:33:40.868,0:33:45.028
to 180, and so this angle[br]here is also Alpha.

0:33:46.410,0:33:48.450
Hang on a minute.

0:33:49.390,0:33:53.702
What we're saying is that the[br]angle that this line makes

0:33:53.702,0:33:58.014
with the tangent is equal to[br]the angle that this line

0:33:58.014,0:33:59.582
makes with the Tangent.

0:34:01.430,0:34:06.904
In other words, if this was a[br]Ray of light coming in here, it

0:34:06.904,0:34:11.205
would be reflected according to[br]the law of reflection, and it

0:34:11.205,0:34:13.160
would pass through the focus.

0:34:14.600,0:34:21.880
So any Ray of light that comes[br]in parallel to the X axis?

0:34:22.710,0:34:26.526
Each Ray of light is reflected[br]to the focus.

0:34:27.240,0:34:32.412
Now you can see how your dish[br]aerial works. Your dish aerial

0:34:32.412,0:34:37.153
is formed by spinning a[br]parabola, so it makes a surface

0:34:37.153,0:34:42.325
a dish. The signal comes in and[br]strikes the dish and is

0:34:42.325,0:34:47.066
reflected to the receptor. That[br]little lump that stands up in

0:34:47.066,0:34:52.669
front of the disk and the whole[br]of the signal is gathered there.

0:34:53.930,0:34:59.292
How else can we might use of[br]this? What we can make use of?

0:34:59.292,0:35:04.271
Its in Searchlight 'cause we can[br]turn it around the other way. If

0:35:04.271,0:35:09.633
we put a bulb there at the focus[br]and it emits light then the

0:35:09.633,0:35:14.229
light that he meets will travel[br]to the parabolic mirror an will

0:35:14.229,0:35:18.442
be reflected outwards in a beam.[br]A concentrated beam, not one

0:35:18.442,0:35:22.655
that spreads but one that is[br]concentrated and is parallel to

0:35:22.655,0:35:24.570
the X axis to the.

0:35:24.620,0:35:25.940
Axis of the mirror.

0:35:26.630,0:35:31.869
So we see that are a property of[br]a curve discovered and known

0:35:31.869,0:35:35.899
about by the ancient Greeks has[br]some very, very modern

0:35:35.899,0:35:40.332
applications, and indeed our SOC[br]would not be the same without

0:35:40.332,0:35:44.362
the kinds of properties that[br]we're talking about now that

0:35:44.362,0:35:46.377
exist in these conic sections.