Find Closest Point on Hyperbola: xy=8 to (3,0)

In summary, the conversation discusses finding the point on the hyperbola xy=8 closest to (3,0). The suggested approach is to use the relation y=8/x to write the distance from (x,y) to (3,0) in terms of one variable. This can then be minimized to find the closest point. There is some confusion about the variables r and t and the term D, as they were not mentioned in the original problem statement.
  • #1
karisrou
15
0
1) Find the point on the hyperbola xy=8 closest to (3,0).

I honestly, have no idea what to do. I seriously do not remember discussing anything like this in class, nor having any previous problems in homework. If anyone can give me a start or walkthrough, that would be fantastic!
 
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  • #2
Well, y = 8/x gives you a relation between the two variables that allows you to write the distance from (x,y) to (3,0) in terms of one variable, which can then be easily minimized.
 
  • #3
I've got D = 2rtx^2 - 6x + 9 + 64/x^2

I'm not sure as to how this minimizes?
 
  • #4
HOW did you get that? What are r and t? What is D? There were not "r", "t", or "D" in your orginal statement of the problem.
 

1. What is a hyperbola?

A hyperbola is a type of mathematical curve that is defined by the equation: xy = 8. It has two branches that open in opposite directions and never intersect.

2. How do you find the closest point on a hyperbola to a given point?

To find the closest point on a hyperbola to a given point, you can use the formula for the distance between a point and a line. In this case, the line is the hyperbola and the given point is (3,0).

3. Can there be more than one closest point on a hyperbola?

Yes, since a hyperbola is a curve, there can be more than one point that is equidistant from a given point. In this case, there can be two closest points on the hyperbola xy=8 to the point (3,0).

4. What is the significance of finding the closest point on a hyperbola?

Finding the closest point on a hyperbola can be useful in many mathematical and scientific applications. For example, it can help determine the shortest distance between two objects or the optimal path for a moving object.

5. Can the closest point on a hyperbola be outside of the hyperbola itself?

Yes, it is possible for the closest point on a hyperbola to be outside of the hyperbola. This can occur if the given point is located outside of the hyperbola's domain or range, or if the hyperbola has multiple branches and the closest point falls on a different branch.

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