This particular video is going to be about of a set of curves
known as conic sections.
Here's a cone.
And conic sections are formed by taking cuts in into this cone in
various ways. Now, the curves that are formed by taking these
particular cuts or sections are very old. They were known to the
ancient Greeks from about 3:50
BC onwards. But not only they very old, but they also a very
modern. The dish aerial that's outside your house that brings
in Sky Television is based upon the reflective property of one
of these conic sections that we're going to be having a look
at Georgia Bank Telescope, a radar telescope that pierces
deep into the universe, picking up signals, recording them for
us again, is based upon a reflective property of one of
these conic sections. So although they are ancient, even
though they are, they have very,
very. Important modern uses.
So let's have a look at our coal. Let's begin by describing
some features about the cone itself, which I'll need in order
to describe the sections and the curves that we're going to see.
First of all, let's think about the axis of the cone. Now,
that's the line that runs from the point down through the
center of the basin out there. So that's the axis.
The generator of the comb. Well, that's any line that runs down
the side of the cone like that and the generator. Think of it
as being a pendulum, a piece of string tight there. Its wings
round here and it Maps out this curved surface of the cone.
So let's begin by looking at the cuts or sections that we can
make. One of the simplest sections that we can make is to
cut across and perpendicular to the axis of the cone. And what
do we get? We get that well known curve circle.
OK, let's not now cut perpendicular. Let's cut at an
angle. So we're going to cut at an angle to the.
Access of the cone.
We're going to keep the cut within the physical dimensions
of the cone. And what do we get? We get a curved surface.
A closed curve again, this time it's an ellipse, kind of
flattened circle if you like, but its name is an ellipse.
Let's put that back on.
And now let's take a cut which is parallel till the
generator parallel to the edge
of the. Curved surface, so there's our edge.
Of the curve surface, and this is the cut that we're going to
take parallel to that edge. So we make the cut.
And there we've got an open curve, this time, not a closed
one. It's an open curve and it's called the parabola.
Putting that back together, what I want you to imagine now is a
double Cole. Now what do I mean by a double cone? I mean another
cone sitting on top of this one.
Point to point. So imagine we've got this cone here. We've got
another cone up here.
And again, I'm going to take a cut, which this time is going to
be parallel to the axis of the coal coming straight down there,
and you can see the cut there now. Once I make it there, I've
made the cut will actually be making the cut through the cone
up here, and so they'll actually be 2 pieces to the curve.
One from the upper cone and one from the lower cone, and that's
a hyperbola. It's got two branches to it.
So those are our three conic sections, sorry, miscounted.
Those are our four conic sections, the circle.
The ellipse.
The parabola. On the hyperbola with
its two branches.
Now the conic sections were first discovered by monarchists
in about 350 BC. Minarchist was an ancient Greek. He in fact had
three different cones, but he took the same cut in each cone
in order to generate three of
them. Here with disregarding the circle, 'cause, That's obviously
the simplest cut to make.
Later Apollonius who was around about 200 BC, generated this
idea of making the cuts all in exactly the same cone.
So. We've seen what these curves look like, and we've
seen how they are generated now. What we want to do is to
explore their mathematical equations.
If we don't explore these mathematical equations, then we
won't be able to deal with these curves and find out what their
properties are. So let's have a look at a definition of these
curves that will actually help us to calculate them.
What I want first of all is a straight line. Straight line has
a name. It's called the direct tricks.
And I want a fixed point. I'm going to label that fixed point
F and it's F4 focus.
Take any point.
In the plane, let's call that the point P.
And then we've got P must be a distance away from this fix
straight line. Let's call that point M and that's of course a
right angle. And it's also a distance away from F. The focus.
And so the definition that we're going to use is that PF.
The distance of the point from
the focus. Is equal to.
A constant multiple of the distance of the point from the
straight line and that constant multiple. We use the letter E to
denote. And so PF is equal to E times by PM.
Now what happens? Well, this point P follows what we call a
locus. Let's just write down that word locus. What do we mean
by a locus? Well, it's simply a path. The Point P follows a path
governed by this rule.
This letter EE is used to stand for what we
call the eccentricity.
Of the curve and we get our different.
Conic sections for different values of E. So if E is
between North and won, the curve that we get is the closed curve.
The ellipse.
If E is equal to 1, the curve that we get.
Is the parabola.
The open curve but with a single
branch. And if he is greater than one, the curve that we get
is the hyperbola. Remember that was the open curve with the two
branches. Now what happens in the rest of the video?
We're going to deal with this particular curve.
In some detail we look at some of its properties and will just
relat say what the equivalent properties are in these two
curves. But this is the one that we're going to deal with in
detail, and the sorts of mathematics that we use and the
ways in which we work if replicated will enable the same
sorts of properties to be proved and established in the ellipse
and the hyperbola. So let's go
on. Have a look at the parabola.
So. Let's draw a picture.
So I'm going to set up some coordinate axes X&Y.
I'm going to take my straight line my direct tricks.
There, and I'm going to take my focus.
There. Now I need some measure, some scale, some size, so I'm
going to say I'm going to take the focus at the point a nought.
Now remember that 4 hour parabola.
PF is equal to PM be'cause E the eccentricity was
equal to 1.
So the distance of our point P has to be the same from.
This line. As it is from this point and of course, there's an
obvious point, namely the origin, that perhaps we'd like
to be on this curve. So that means that the direct tricks
here is going to be at X equals
minus a. Let's put our.
Point P. Let's say there.
There is.
PM That's our point. N there is PF
will call this the point XY.
So what's the locus? What path? What's the equation of that path
that P is going to follow as it moves according to this
definition, one of the things we better do is we better write
down what are these lens?
PM equals well from M to the Y axis is a distance A and
there's a further distance of X to go before we get to pee.
So PM is A plus X.
What about PF?
Once we look at PF, it's the hypotenuse of a right angle
triangle which we can form by dropping a perpendicular down
there so we can see that PF.
Will be when we square it. PF squared will be equal to
that squared, which is just the height, why?
Plus that squared.
Will up to there is X and up to there is a so that little bit in
there is a minus X.
Squared
So this is PF squared.
Let's just check that again, PF is the hypotenuse of a right
angle triangle, so using Pythagoras PF squared is that
squared plus that squared. That's the height YP is above
the X axis.
Up to there is X up to. There is a, so the distance between there
and there is a minus X.
So now I need to equate these two expressions and first of all
that means I've got to square PM because if PF is equal to PMPF
squared. Must be equal to PM squared and so we can substitute
this in so instead of PF we have Y squared A minus, X all
squared and instead of PM squared we have a plus X all
squared. Now we need to multiply out this bracket Y
squared plus. Now let's just do the multiplication over here.
A minor sex, all squared is a minus X times by A minus X.
So we've a Times by a that gives us a squared.
With a Times by minus X and minus X times by a, which gives
us minus two AX and then we have minus X times Y minus X which
gives us plus X squared equals. Let's have a look at this
bracket. It's a plus X all squared, so it's going to be
exactly the same as this one, except with plus signs in. So
we're going to get a squared +2.
X plus X squared.
Now let's have a look at this. We've A plus a squared here and
a plus a squared there so we can take an A squared away from each
side. We have a plus X squared here under plus X squared here
so we can take an X squared away
from each side. We've minus two X here and plus 2X there, so it
makes sense to get those axis together by adding 2X to this
side and adding it to that side.
So if we do all that we've got Y squared.
The two A squared will disappear, subtracting a square
from each side. The two X squared will disappear.
Subtracting X squared from each side and adding the two X2 each
side, we get 4A X that is the standard Cartesian equation
for a parabola.
Now.
This is a standard equation and it is the equation with which we
want to work. But sometimes when we're doing these
curves, it's helpful to have the equation described in
terms of 1/3 variable.
Variable that's often called a parameter.
And the parameter we're going to have in this case is the
parameter. T.
So we've got an equation Y squared equals 4A X.
And what we're looking for is a way of expressing X in terms of
tea and a way of expressing Y in terms of T.
When we look at this, this says Y squared equals 4A X. Now Y
squared is a complete square. It's a whole square an exact
square. I can take it square root and I'll just get plus
online as why so? Can I do it over here? Can I choose an
expression for X that will give me a complete square?
On this side of the equation, well, four is already a complete
square. If I want to make a a complete square and have to have
a squared. So I need something in X that's gotten a attached to
it. So 8 times by a would give me a squared.
And then I want to be able to take this exact square root.
Will the only thing if I'm going to introduce this
variable T that suggests itself and he sensible is T
squared? So if we put.
X equals AT squared, then Y squared is
equal to four a Times 80 squared, which
is 4A squared T squared. And so why
is equal to two
AT? And So what I've got here is what's called
the parametric equation.
For the parabola.
Now that's our parametric
equation. And this is our Cartesian Equation. One thing we
haven't done yet is sketched.
The curve itself, so just want to do that.
So there's a.
X&Y axis.
There's our direct tricks through minus a.
Here's our focus at a nought.
We know the curve is going to go through there and it's equation
is Y squared equals 4A X. Well, if we take positive values of X,
we can see we're going to get that, but a positive value of X
gives us the value of Y, which when we take the square root is
plus or minus. So we can see we're going to have that and.
Symmetry in the X axis. Notice we can't have any negative
values of X because 4A A is a positive number times a negative
value of X would give us a negative value for Y squared.
What about the parametric equation X equals 80 squared Y
equals 280? What does this do in terms of this picture? Well, if
we look at it, we can see X equals 80 squared. A is a
positive number, T squared is positive, so again we've only
got values of X which are greater than or equal to 0.
However, values of why can range from minus Infinity to plus
Infinity? And if you like tea down here is minus Infinity.
Going round to T equals 0 here at the origin coming round. If
you like to T is plus Infinity
there. So T is a parameter and in a sense it counts is around
the curve as T increases from minus Infinity to plus Infinity,
we move around the curve.
OK. What about this curve? Well, one of
the things that we want to be able to establish is what's the
equation of the tangent to the curve at a particular point.
So again, let's draw our picture of the curve, putting
in the direct tricks straight line, putting in the focus
and putting in.
The curve.
Let's take any point P on the curve.
And the question we are asking is what is the equation of the
tangent to the curve?
What's the equation of that lie? Well, I've got P is any point on
the curve. So instead of calling XY, I'm going to use the
parametric equation. X is 80 squared, Y is 280.
So X is 80 squared, Y is 2 AT.
The question we're asking what is the equation of the tangent?
Well, it's a straight line to find the equation of a straight
line, we need a point on that line or we've got it 80 squared
280 and we need the gradient of the line and the gradient of the
line. Of course must be the gradient of the curve at that
point P, so we need DY by X.
But both X&Y are defined in terms of T.
They define in terms of another variable their functions of
another variable. So if I use function of a function DY by DT
times DT by The X.
Then I can calculate the
gradient. Divide by DT and let's remember that DT by DX
is the same as one over the X by DT. In other
words, D why by DX divided by the X by DT.
Divide by DT is the derivative of this.
The xpi DT is going to be the derivative of that.
Divide by DT is just to
A. Divided by the derivative of X and the derivative
of that is 2 AT.
Which gives us a gradient of one
over T. So now we've got the gradient curve at that
point, so we've got the gradient of the straight line, and we've
got the point, so Y minus Y one over X Minus X one is equal to
the gradient, so that's a standard formula for finding the
equation of a straight line given a point on the line X1Y
one and it's gradient. So let's substitute in the things that we
know. This is going to be our X
one. This is going to be our why one because that's the point on
the curve. This is going to be our value of M our gradient, so
let's just write down that information again. So X one we
said was going to be the point on the curve X equals 80
squared. Our Y one was going to be to AT and our gradient
M, which was the why by DX was going to be one over T.
The standard equation of a straight line given a point
on the line X1Y one and it's gradient M. Now let's
substitute that information in so we have Y minus 280.
All over X minus 8 Y squared is equal to one over
T. Now we need to multiply everything by T in order to get
tea out of the denominator here and multiply everything by X
minus 8 Y squared in order to get the X minus 8 Y squared out
of the denominator. So and multiply everything by T so we
have T times Y minus 280 is equal to and we multiply
everything by X minus 8 Y squared and on this side it
means we're multiplying by.
One, so that's just X minus 8 Y squared. Multiply out this
bracket. Ty minus 280 squared is equal to X minus AT squared.
Now. Here we've got 80 squared term in 80 squared minus 280
squared. And here we've got another term in 80 squared which
is just minus 80 squared. So if I add 280 squared to each side,
I'll just have 80 squared on this side and this will just
leave me with T. Why? So that's Ty equals X plus 80 squared.
And that's the equation of the tangent at the point P.
Now there is a reason for working out this tangent. It's
not just an exercise in using some calculus and using some
coordinate geometry. I've done it for a purpose. I want to
explore the reflective property of a parabola.
So first of all, let's think what that actually
is reflection.
If we have a flat plane surface, then if a beam of light comes
into that surface at an angle theater, the law of reflection
tells us that it's reflected out again at exactly the same angle
to the surface. So it comes in it's reflected out again.
What if we had not a plain flat surface?
But let's say a.
Curved surface well what the law says is, I'm sorry, but a plane
for a curved surface is exactly the same as a plain flat
surface, except we take the tangent at the point where it's
coming in. So the Ray of light is reflected out at the same
angle. To the tangent that it made to the tangent when it came
in. So that's the reflective
law. How does this affect a parabola? Say we had a
parabolic mirror mirror that was in the shape of our parabola.
As our parabola, what happens to a Ray of light that's a comes
in parallel to the X axis, and it strikes the mirror when it
comes in, it must be reflected.
Question is. Where is it reflected? What direction does
that going? The law of reflection tells us.
That if that's the angle theater that's made there,
then that must be the angle theater that's made there.
So what we need to discover is.
If this is the case, where does that go? Where is that Ray of
light directed? And as you can see, this is all to do with the
tangent to the curve, so it's quite important that we know
what the equation of that particular tangent is.
OK.
Now we know what it is we're trying to do. Let's set up a
diagram. And on this diagram, let's put some.
Coordinates of important things, so there's the direct tricks,
and there's our focus. 2 very
important points. Here's our
parabola. Whoops, a Daisy.
Better curve, here's our point, P.
There and here's our tangent to the point P.
I'll call that the point T there an array of light.
Going to come in like so I'm going to do. I'm going to join
that to the focus.
So here I've got this line here, which is parallel.
To the X axis I'm going to call this end, so I've got a line
here, PN. If I extend PN back in a straight line, it's going to
meet the direct tricks there. At the point M and there will be a
right angle. But remember what the
point Pierce .80 squared to AT.
OK.
One of the things that we need to find is this point T down
here, let's remember the equation of our tangent. It's Ty
is equal to X plus 80 squared. That's the equation of the
Tangent. PT. So
at T. Y is equal to 0.
That implies that X is equal to minus AT.
Square.
That enables us to find this length here
TF. Must be that distance up to the origin. Oh, which
is a distance 80 squared plus that. So that's a plus
AT squared.
We know that PF is equal to P.
N.
Because that comes from the definition of the parabola.
But we also know what PM actually is. PM is
this distance A plus? This distance here, which is 80
squared. So if we look at the argument that we've got here,
that TF is A plus 80 squared PM is A plus. 8880 squared and
PM is equal to PF. And this means that TF and PF have to
be the same length.
In other words, this triangle here.
Is an isosceles triangle.
Triangle.
FTP
is.
Isosceles OK, it's isosceles. What does
that tell us? Well, if it's isosceles, it tells us that it's
two base angles are equal, so that's one equal side, and
that's one equal side, so that angle has to be equal to that
angle there. Therefore, the angle FTP is equal to the angle
TPF. Let me just mark those that
angle there. Is Alpha.
And that angle there is Alpha.
But wait a minute.
This line and this line are
parallel. So therefore, this angle NPF is equal
to this angle PF.
T. Those two angles are both equal to.
Beta.
Be cause this line is parallel to that one, they make a zed
angle. Now the things of this triangle add up to 180 and
here is a straight line, the angles of which must add up
to 180, and so this angle here is also Alpha.
Hang on a minute.
What we're saying is that the angle that this line makes
with the tangent is equal to the angle that this line
makes with the Tangent.
In other words, if this was a Ray of light coming in here, it
would be reflected according to the law of reflection, and it
would pass through the focus.
So any Ray of light that comes in parallel to the X axis?
Each Ray of light is reflected to the focus.
Now you can see how your dish aerial works. Your dish aerial
is formed by spinning a parabola, so it makes a surface
a dish. The signal comes in and strikes the dish and is
reflected to the receptor. That little lump that stands up in
front of the disk and the whole of the signal is gathered there.
How else can we might use of this? What we can make use of?
Its in Searchlight 'cause we can turn it around the other way. If
we put a bulb there at the focus and it emits light then the
light that he meets will travel to the parabolic mirror an will
be reflected outwards in a beam. A concentrated beam, not one
that spreads but one that is concentrated and is parallel to
the X axis to the.
Axis of the mirror.
So we see that are a property of a curve discovered and known
about by the ancient Greeks has some very, very modern
applications, and indeed our SOC would not be the same without
the kinds of properties that we're talking about now that
exist in these conic sections.