Identifying if a parabola exists from given key features
About this video
The video asks whether a parabola can have focus F(-8, 2), directrix y = -6 and pass through P(6, 14). Using the definition that every point on a parabola is equally far from the focus and the directrix, it finds the distance from P to the directrix is 20 but the distance from P to F is sqrt(340), which is not 20, so no such parabola exists.
You'll be able to
- state the focus-directrix definition of a parabola.
- find the distance from a point to a horizontal line.
- use the distance formula to find the distance between two points.
- decide whether a point can lie on a parabola with a given focus and directrix.
Key words
- parabola
- The set of all points that are the same distance from a fixed point (focus) and a fixed line (directrix).
- focus
- The fixed point used to define a parabola.
- directrix
- The fixed line used to define a parabola.
- locus
- The set of all points that satisfy a condition (plural: loci).
- distance formula
- d = sqrt((x2 - x1)^2 + (y2 - y1)^2), which comes from the Pythagorean theorem.
Transcript
- [Instructor] We're asked, can a parabola exist with the point P at 6, 14, a focus at point F, I guess, which is at -8, 2, and a directrix at y = -6? Why or why not? So, pause this video, have a go at that, and then we'll do it together. All right, let me just try to sketch this a little bit just to visualize what's going on.
So, they're saying that we have to figure out whether this point could be on a parabola with this focus in that directrix. See, the focus is all the way at -8, 2, and then the directrix is at y = -6. And then this point is all the way at 6, 14, so let me actually let me just draw the whole or try to draw a fairly large coordinate plane here.
Y axis, X axis, and then let me go... We just have to sketch it, so let's say this is 5, 10, 15,
- 5, -10, -15. Let me just do it like this way. I don't have to go that low.
- 5, -10. - 5, -10. And then let's make this 5 and 10. And then last but not least, -5, -10, and -15. All right, so the point is at 6, 14, so I am at 6, 14, which is right about there. I will call that P. The focus, -8, 2.
- 8, 2. So, that would put us right about there. It's actually quite closer. Right around there. So, this is the focus. And directrix at y = -6, so y = -6 is right about here. And so, that is right over there. y = -6. Now, our definition of a parabola, it's the loci or loci, I always forget how to pronounce that, I think loci, of all of the points that are equidistant from the focus and the directrix.
And so, what we need to do is, if this is on the parabola, then the distance from P to F, this distance right over here, needs to be equal to the distance between P and this directrix. So, let's figure out what each of these... Actually, I could just do this one by going...
Whoops, by just going straight down. So, this distance right over here, this vertical distance is pretty easy to figure out because this P right over here is at 6, 14, that's at y = 14. The directrix is at y = -6, so 14 + 6. This distance over here is 20. Now, what's the distance between P and F?
Well, the distance formula, which comes out of the Pythagorean theorem. So, the distance is going to be equal to the square root of the difference between the X coordinates squared plus the difference between the Y coordinates squared, so I could say 6 minus negative, let me be careful, 6 minus -8 squared plus 14 minus 2 squared.
Well, let's see, this is going to be 6 + 18, 6 + 8. I'm over. I just had a fever, so my brain's a little bit foggy still. So, 6 + 8 is 14 squared, and then this is going to be plus 12 squared. So, let's see, this is going to be, the square root of 14 squared is 196 + 144.
And so, what is that going to be? That's going to be... Am I doing this right? Yep, that's gonna be the square root of 340. And we already know that this value is not equal to 20. This does not equal 20. 20 squared is 400. So, we already see that this distance and that distance are not equal.
And so, this point cannot exist on the parabola that has that as a focus and that as a directrix.
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