WOMAN: So in this interactive, we investigate graphs of conics
which have a focus at the point (2, 0)
and a directrix on the line x = 4,
and we'll look at the different conics we can create
by adjusting the eccentricity.
So we first consider the definition of eccentricity
and the eccentricity is to do with the ratio
of the distance from a point on the locus to the focus
when compared with the distance from a point on the locus
to the directrix.
And so if S is the focus and P is a point on the locus
and D is a point on the line that is perpendicular to P,
then the eccentricity is calculated
by the length of PS divided by the length of the line PD.
So in this particular diagram here, we have ...
... clearly we'll have an eccentricity that is greater than 1
since PS is larger than PD.
So returning to our interactive,
we note that what we can adjust here is the eccentricity,
so we're looking at what happens as we change the eccentricity
given this particular focus and this particular directrix.
So let's just run through it,
and we see that as we increase the eccentricity,
we're first starting with an ellipse that's getting larger.
We then create a parabola ...
... which then moves into a hyperbola.
So we note that we're getting ...
When we have an eccentricity greater than 1, we get a hyperbola,
when we have an eccentricity equal to 1
we get the specific case of a parabola,
and when the eccentricity is less than 1
we're getting the ellipses.
So let's just have a little look at this a bit further.
So here we see the same set-up that we're looking at in the interactive -
a directrix at x = 4, a focus at the point (2, 0) -
and we're looking at the specific case where the eccentricity is 1,
so where we're looking at the locus of all the points
that are equidistant from the focus and from the directrix.
And we can see here, by moving this point, we can see that illustrated.
So here we're seeing that this point here on the locus
is, in fact, equidistant from our focus
and from our directrix,
and as we move the point around the graph
we see that we are still maintaining
the locus of all of the points
where the distance from the locus to the focus
is equal to the distance from the locus to the directrix,
and this will create a parabolic shape.
And here now, we're looking at the case where the eccentricity is 0.6,
so given that it's less than 1, we're seeing an ellipse.
So an eccentricity of 0.6
means that the distance from the locus to the focus
is 0.6 of the distance from the locus to the directrix,
and we can see that illustrated here.
So as I move this point, we should be able to see that being maintained.
So this distance here between the focus and the locus
is about 0.6, so just a little over half the distance
from this point here to the directrix out here,
and that's going to be the same for every point around this ellipse.
And the third case now is where the eccentricity is greater than 1.
So we'll consider the case where the eccentricity is 2,
so what that means is that the distance
from a point on the locus to the focus
is twice as big as the distance
from a point on the locus to the directrix.
And we can see that that is indeed the case here,
and as we move this point
that distance to the focus continues to be twice as big
as the distance to the directrix,
and that would be true for every single point on this hyperbola.
And so this interactive allows us to explore
those properties of these conics
with a focus at (2, 0) and a directrix at x = 4,
so we note that an eccentricity less than 1 gives an ellipse,
an eccentricity equal to 1 gives a parabola,
and an eccentricity greater than 1 gives a hyperbola.