www.mathcentre.ac.uk/.../Conic%20sections.mp4

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.