Text description - screencast for interactive 3

[In this video, an electronic x and y graph is used to show a conic. Below the graph, an adjustable slider allows individual values in the equation to be changed. As the values in the equation are changed, the conic forms and changes shape. The narrator describes the changes in the graph.]

NARRATOR: 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 equals 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.

[On an x and y graph, a vertical line runs from point D down, crossing the x-axis. From a point marked S on the x-axis, a slightly angled line runs up to a point marked P, which is at the same level as D. A third line joining P and D runs parallel to the x-axis and forms a right angle at D with the first line.]

NARRATOR: 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.

[An x and y graph features both the SDP diagram and the parabola which is open to the left. The parabola passes through point P of the diagram. As point P is moved along the parabola, point D moves up and down the vertical line and the diagram's shape changes.]

NARRATOR: So here we see the same set-up that we're looking at in the interactive — a directrix at x equals 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.

[The parabola becomes an ellipse.]

NARRATOR: 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.

[A hyperbola appears on either side of the diagram's vertical line.]

NARRATOR: 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.

[The interactive graph appears onscreen.]

NARRATOR: And so this interactive allows us to explore those properties of these conics with a focus at (2, 0) and a directrix at x equals 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.