Identifying if an ellipse or hyperbola exists from given key features
When writing the hyperbola condition the narrator says 'plus d(P2, F2)' where it should be a difference (minus). The later calculation correctly uses the difference.
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With foci F1(1, 0) and F2(-1, 0) and points P1(0, 1) and P2(0, -1), the video checks whether the points could lie on an ellipse (equal sums of distances to the foci) or a hyperbola (equal absolute differences). Every distance works out to sqrt(2), so the sums are both 2 sqrt(2) and the differences are both 0, meaning the points could lie on an ellipse or a hyperbola.
You'll be able to
- state the focus-based definitions of an ellipse and a hyperbola.
- use the distance formula to find distances from points to foci.
- decide whether given points could lie on an ellipse, a hyperbola, or both.
Key words
- foci
- The two fixed points (plural of focus) used to define an ellipse or hyperbola.
- ellipse
- All points whose distances to the two foci add up to the same constant.
- hyperbola
- All points where the absolute difference of the distances to the two foci is the same constant.
- absolute value
- The size of a number without its sign, so it is never negative.
- conic section
- A curve made by slicing a cone, such as a circle, ellipse, parabola or hyperbola.
Transcript
- [Instructor] We are asked is a conic section with foci at F sub one, and they give the coordinates, and F sub two, and these coordinates, and points at P1 and P2 with these corresponding coordinates, an ellipse, a hyperbola or both? Why or why not? Pause this video, have a go at that before we do this together.
All right, so let's just remind ourselves, if we're dealing with an ellipse, then we know that the distance from P1 to F1 plus the distance from P1 to F2, so the sum of the distances of P1 to the different foci, that should be equal to the sum of the distances for point two.
So from P2 to F1 and from P2 to F2. Now, if we're dealing with a hyperbola, so let me write that over here. So the Hyperbola case. Hyperbola, instead of a sum, we're looking at the difference and we're looking at the absolute value of the difference of the distances.
So in the hyperbola situation, we're gonna have the distance from P1 to F1 minus the distance from P1 to F2. And we just care about the absolute value of it. We want this to be positive. That needs to be equal to the absolute value of the differences of the distances from P2 to the foci.
So plus d(P2, F2) and we're done. My hand is hurting with all that notation. So what we have to do is just figure out these things and we don't have to figure out eight things, we only have to figure out four of these and then we can look at 'em in each of these cases.
So first, let's look at the distance from P1 to F1. So that's going to be, I'll just do it right over here, it's going to be the square root of our difference in x values. So it's going to be zero minus one squared. This is just the distance formula. And then plus the difference in y value.
So one minus zero squared, one minus zero squared. Well, this is just going to be one plus one, square root. So this is just going to be the square root of two. So that's the square root of two. And by the way, this is also right over here, the square root of two.
And now this one over here, distance from P1 to F2. Well, same idea. Let's see, from P1 to F2. So we have zero minus negative one, zero minus negative one squared, and then we have plus one minus zero squared, one minus zero squared. Well, this is just going to be one, this is going to be one squared.
Well, one squared plus one squared is just two. And so this is once again going to be square root of two. So the left-hand side over here has added up to two times the square root of two. And in this situation down here, this was the square root of two. So the left-hand side, the difference, the absolute value of the difference over here is just going to be zero.
Now let's do the right-hand sides, and I'm going to find more colors. So the square root of P2, zero, one, so zero minus one squared plus one, sorry, plus negative one minus zero squared, negative one, negative one minus zero squared. Well, once again, this is one, this is one.
You square 'em, you still get ones, you add 'em, you get a two. So this is going to be square root of two. This is also right over here, square root of two. And then last but not least, this piece right over here, I'm sensing a pattern. So we see, let's see, we're going from zero to negative one.
So zero minus negative one squared plus negative one minus zero squared. So same idea, this all simplifies to square root of two. So this is also square root of two. So in this top scenario, this is two square roots of two. So the sum of the distances from each of the points to the two foci, they're actually the same.
So these two points could be on an ellipse, but we also see the difference of the distances from each point to the foci. It stays constant. So they could also be on a hyperbola. So this is a case where the same set of two points could be on an ellipse or a hyperbola.
And so you could imagine if you were to construct an ellipse that has those two points in a hyperbole that had those two points, those would be two points where those two conic sections would actually intersect.
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