Identifying the foci of an ellipse using the radii
About this video
For an ellipse centred at (5, -2) with radii 6.5 (vertical, to (5, 4.5)) and 2.5 (horizontal, to (7.5, -2)), the video finds the focal length using f = sqrt(p^2 - q^2) = sqrt(6.5^2 - 2.5^2) = 6. Since the foci lie on the major axis, it moves 6 up and down from the centre to get foci at (5, 4) and (5, -8).
You'll be able to
- read the major and minor radii of an ellipse from its centre and key points.
- use f = sqrt(p^2 - q^2) to find the focal length.
- locate the foci by moving the focal length along the major axis from the centre.
Key words
- foci
- The two special points inside an ellipse, lying on the major axis.
- focal length
- The distance from the centre of an ellipse to each focus.
- major axis
- The longer axis of an ellipse; the foci lie on it.
- minor axis
- The shorter axis of an ellipse, perpendicular to the major axis.
Transcript
- [Instructor] So what we wanna do is figure out the foci for this ellipse. And so pause this video, have a go at this before we do this together. All right, so this all boils down to figuring out what the focal length is and we can figure out the focal length based on the lengths of the major and minor axes.
If F is the focal length, we know that the focal length is going to be equal to the square root of the length of the major axis squared minus the length of the minor axis squared. And we proved this in other videos. It really comes straight out of the definition of what an ellipse is.
So to figure out F, let's figure out P and Q first. So what is the length of the major axis? So the center here is at five, negative two, and we have this point here at five, 4.5. So let's see, this length is, looks like it's 6.5, right? We have two here, and then another 4.5.
So this is 6.5 squared minus. And then the length of the minor axis, we're going from five to 7.5. So this is 2.5 right over here. So minus 2.5 squared. And then we're going to take the square root of all of that to figure out the focal length. So let me get the calculator out.
I have 6.5 squared minus 2.5 squared is equal to, and then you take the square root to that, you just get six. So our focal length is equal to six, but they're not asking us the focal length or I didn't ask you the focal length. We need to figure out where are the foci.
So the foci sit on the major axis. One way to think about that. And if we go six above this point, right over here, we're at negative two. If we go six above, we'll be right over here. So this one foci will be at the point five and then negative two plus six is four, and then the other would be at five.
And then one, two, three, four, five, six. Five. And then negative two minus six is negative eight. And we're done, we have identified the foci for this ellipse.
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