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WOMAN: So in this interactive,
we investigate graphs of conics

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which have a focus
at the point (2, 0)

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and a directrix on the line x = 4,

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and we'll look at the different
conics we can create

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by adjusting the eccentricity.

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So we first consider
the definition of eccentricity

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and the eccentricity
is to do with the ratio

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of the distance from a point
on the locus to the focus

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when compared with the distance
from a point on the locus

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to the directrix.

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And so if S is the focus
and P is a point on the locus

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and D is a point on the line
that is perpendicular to P,

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then the eccentricity is calculated

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by the length of PS divided
by the length of the line PD.

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So in this particular diagram here,
we have ...

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... clearly we'll have an eccentricity
that is greater than 1

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since PS is larger than PD.

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So returning to our interactive,

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we note that what we can adjust
here is the eccentricity,

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so we're looking at what happens
as we change the eccentricity

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given this particular focus
and this particular directrix.

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So let's just run through it,

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and we see that as we increase
the eccentricity,

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we're first starting with an ellipse
that's getting larger.

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We then create a parabola ...

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... which then moves into a hyperbola.

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So we note that we're getting ...

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When we have an eccentricity
greater than 1, we get a hyperbola,

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when we have
an eccentricity equal to 1

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we get the specific case
of a parabola,

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and when the eccentricity
is less than 1

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we're getting the ellipses.

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So let's just have a little look
at this a bit further.

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So here we see the same set-up that
we're looking at in the interactive -

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a directrix at x = 4,
a focus at the point (2, 0) -

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and we're looking at the specific
case where the eccentricity is 1,

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so where we're looking
at the locus of all the points

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that are equidistant from the focus
and from the directrix.

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And we can see here, by moving this
point, we can see that illustrated.

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So here we're seeing
that this point here on the locus

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is, in fact, equidistant
from our focus

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and from our directrix,

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and as we move the point
around the graph

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we see that we are still maintaining

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the locus of all of the points

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where the distance
from the locus to the focus

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is equal to the distance
from the locus to the directrix,

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and this will create
a parabolic shape.

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And here now, we're looking at the
case where the eccentricity is 0.6,

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so given that it's less than 1,
we're seeing an ellipse.

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So an eccentricity of 0.6

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means that the distance
from the locus to the focus

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is 0.6 of the distance
from the locus to the directrix,

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and we can see that illustrated here.

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So as I move this point, we should
be able to see that being maintained.

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So this distance here
between the focus and the locus

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is about 0.6, so just a little
over half the distance

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from this point here
to the directrix out here,

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and that's going to be the same
for every point around this ellipse.

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And the third case now is where
the eccentricity is greater than 1.

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So we'll consider the case
where the eccentricity is 2,

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so what that means
is that the distance

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from a point on the locus
to the focus

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is twice as big as the distance

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from a point on the locus
to the directrix.

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And we can see that that
is indeed the case here,

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and as we move this point

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that distance to the focus
continues to be twice as big

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as the distance to the directrix,

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and that would be true for every
single point on this hyperbola.

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And so this interactive
allows us to explore

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those properties of these conics

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with a focus at (2, 0)
and a directrix at x = 4,

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so we note that an eccentricity
less than 1 gives an ellipse,

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an eccentricity equal to 1
gives a parabola,

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and an eccentricity
greater than 1 gives a hyperbola.