Common tangent of circle & hyperbola (1 of 5) (video) | Khan Academy
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So this is a circle. This right here is a circle with center at x equals 4. At x equals 4, y is equal to 0.
Three points defining a circle
And it has a radius of 4 as well. So let me graph this circle here. So let me draw the horizontal axis, my x-axis. Let me draw the y-axis. That is my y-axis over here. And let me draw its center, so 1, 2, 3, 4. That's it's center, and it has a radius of 4. So it's going to come out, and it's going to look something-- I could draw a better circle than that. It's going to look something like-- that's the top half, and then the bottom half is going to look something like that.
So that's our circle.
Meet Me In The Circle | Our World In Ruins
Now let's think about the hyperbola. So if we just look at it, the x squared term is positive, so it's going to be a hyperbola that opens to the right and the left. We do this a bunch in the conic sections videos if you want to review of that. And we could just figure out where it intersects the x-axis. So then when y is equal to 0, we have x squared over 9 must be equal to 1. Or x would be plus or minus 3.
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- Common tangent of circle & hyperbola (1 of 5)
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So the hyperbola is going to look like this. So this is at plus 3 comma 0. The hyperbola will open up like that. And then at 1, 2, 3, negative 3 comma 0, the hyperbola is going to open up to the left. And so in the problem when they describe the points A and B, they're probably talking about that point A and that point B.
Now, let's think about what this question is asking us. Equation of a common tangent with positive slope-- so it has to have a positive slope-- to the circle as well as to the hyperbola-- a common tangent.
So let's just think about this a little bit. So it's going to have a positive slope, so it won't be tangent to the circle anywhere where the circle has a negative slope. So it can't be tangent over here.
It can't be tangent over there. And then we could say, well, if it was tangent to the circle over here, what would happen?
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Wellit wouldn't be able to be tangent to the hyperbola. So it has to be tangent to the circle someplace in this blue region right over here. And then how can it be tangent to the hyperbola?
It might be tempting to say that it would be tangent to the hyperbola in this way somehow, but what you need to realize is the hyperbola is asymptoting towards some line. And we could figure out what that line is. Its asymptoting towards some line.
So the way can we can find it is we draw a perpendicular bisector of each of these sides and where the three perpendicular bisectors intersect-- and we show that they always intersect at a unique point-- that is that circumcenter.
And I'll do it really quick right over here. So let's say that this is the perpendicular bisector of that side, this is the perpendicular bisector of that side, and this is the perpendicular bisector of that side.
So these are all perpendicular, this is perpendicular, and they each bisect the sides.
B to this point is going to be equal to this point to A. A to this point is going to be equal to that point to C. C to this point is going to be equal to that point to B. And this point right over here, we've already talked about, we'll call that point O. We call that the circumcenter. O is the circumcenter. This is all a little bit of review. So if you have three points, you have a unique triangle.
That unique triangle has a unique circumcenter, which is equidistant to the three points of the triangle, three-- I should say the three vertices of the triangle-- and that distance between the circumcenter and the three points, the three vertices, I should say. So let me draw that in a different color. And we've learned when we first talked about circles, if you give me a point, and if we find the locus of all points that are equidistant from that point, then that is a circle.
And when I say a locus, all I mean is, the set of all points. If you give me any point right over here, so that's an arbitrary point, and you also specify a radius, and say what is the set of all the points on this two dimensional plane that are equidistant, that are that radius away from the center? It uniquely defines a circle. That's how we defined a circle right over here. And similarly, if you say, look, if you start with the center at O, and you say all of the points that are the circumradius away from O, it will uniquely identify a circle.
And that circle will contain the points A, B, and C because those are the circumradius away from O. So they are included in that set. So the circle would look something like-- let me draw it. It would look something like this-- trying my best to draw it, just like that. Everything we've talked about, just now within the last few minutes, is all review.
We know all of this. But I went over it just to kind of reinstate a pretty interesting idea, that if you give me three points that defines a unique triangle, and if you have a unique triangle-- And let me make it clear. This is three non-collinear points, so three points not on the same line. If you have three points that are not on the same line, that defines a unique triangle.
For any unique triangle you have a unique circumcenter and circumradius. I'll rewrite it, I don't want to get lazy and confuse you-- circumradius.
And if you give me any point in space, any unique point, and a radius, the set of all points that are exactly that radius away from it, that defines a unique circle. So we went through all of this business of talking about the unique triangle, and the unique circumcenter, and the unique radius, to really just show you that if you give me any three points that eventually, really, just defines a unique circle.
So just as you need three points to define a triangle, you also need three points to define a circle, two points won't do it. And one way to think about it is, if you give me two points, there's an infinite number of triangles that I construct with those two points, because I could put the third point anywhere.
I could construct this triangle. I could construct this triangle, I could construct this triangle, I can construct this triangle. And all of these triangles are going to have different circumcenters and different radiuses. And so they're going to have different circles that circumscribe about those triangles.
So this one-- so for example, this would be one circle that could go around, that could circumscribe that triangle. You could have this circle right over here. So you see clearly, very clearly, that two points are not enough. You need three points, three points lead to a triangle, lead to a unique circle.