The slice must be steeper than that for a parabola, but does not have to be parallel to the cone's axis for the hyperbola to be symmetrical. By placing a hyperbola on an x-y graph centered over the x-axis and y-axis , the equation of the curve is:.
Any branch of a hyperbola can also be defined as a curve where the distances of any point from:. So this Cartesian framework is a very powerful tool to allow us to extract properties of the hyperbola to prove things and to apply it in a lot of different situations. Very nice, clean, algebraic form for a beautiful geometrical object. Share this post. The hyperbola is a rather special conic section. It is the curve that we get when we slice a cone with a plane that meets both the top and bottom of the cone.
So a hyperbola has two distinct branches. It also has associated to it two special lines called asymptotes , which it approaches as we move away from the centre. Want to keep learning? See other articles from this course.
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Create an account to receive our newsletter, course recommendations and promotions. Like an ellipse , an hyperbola has two foci and two vertices; unlike an ellipse, the foci in an hyperbola are further from the hyperbola's center than are its vertices:.
The hyperbola is centered on a point h , k , which is the " center " of the hyperbola. The point on each branch closest to the center is that branch's " vertex ". The vertices are some fixed distance a from the center. The line going from one vertex, through the center, and ending at the other vertex is called the "transverse" axis. The " foci " of an hyperbola are "inside" each branch, and each focus is located some fixed distance c from the center.
The values of a and c will vary from one hyperbola to another, but they will be fixed values for any given hyperbola. For any point on an ellipse, the sum of the distances from that point to each of the foci is some fixed value; for any point on an hyperbola, it's the difference of the distances from the two foci that is fixed.
This fixed-difference property can used for determining locations: If two beacons are placed in known and fixed positions, the difference in the times at which their signals are received by, say, a ship at sea can tell the crew where they are. Death of a Salesman Dr. Jekyll and Mr. SparkTeach Teacher's Handbook. Summary Hyperbolas.
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