Tangents to Curves: Intersection in First Quadrant

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Homework Help Overview

The discussion revolves around the tangents to the curves defined by the equations xy=1 and x^2 - y^2 = 1, specifically at their points of intersection in the first quadrant. Participants are exploring the nature of these tangents and their relationships to the curves.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to find the points of intersection of the curves and are discussing the derivatives of the functions. Questions are raised about the significance of the tangents at the intersection points and the behavior of the curves in the first quadrant.

Discussion Status

There is ongoing exploration regarding the intersection points of the curves and the tangents. Some participants express uncertainty about the existence of intersections, while others suggest that there are indeed two points of intersection. Guidance is provided on how to approach solving the equations simultaneously.

Contextual Notes

Participants note that they are working under the constraints of the problem statement, which specifies the first quadrant and the need to find intersections. There is also mention of the derivatives and their implications without reaching a final conclusion.

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Homework Statement



Is there anything special about the tangents to the curves xy=1 and (x^2) - (y^2) =1 at their point of intersection in the first quadrant.

The Attempt at a Solution



I know what the derivatives of both functions are and what they look like when graphed. But, I'm not sure what to do with it beyond that. For the first function I have that dy/dx=-1/x^2 and for the second I have dy/dx = x/sqrt(-1+x^2). What is different about these functions and what happens in the first quadrant?
 
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mstudent123 said:

Homework Statement



Is there anything special about the tangents to the curves xy=1 and (x^2) - (y^2) =1 at their point of intersection in the first quadrant.

The Attempt at a Solution



I know what the derivatives of both functions are and what they look like when graphed. But, I'm not sure what to do with it beyond that. For the first function I have that dy/dx=-1/x^2 and for the second I have dy/dx = x/sqrt(-1+x^2). What is different about these functions and what happens in the first quadrant?

Well, according to the problem statement, the functions intersect there. Seems to me that it would be useful to find the point of intersection.
 


I have been trying to do that. However, I have not been able to find the intersection of the tangents and there is not intersection of the original functions.
 


mstudent123 said:
I have been trying to do that. However, I have not been able to find the intersection of the tangents and there is not intersection of the original functions.

There are in fact two points of intersection. Why do you think there are none?

You need to find (x,y) that satisfies both xy = 1 and x^2 - y^2 = 1. How do you solve two equations with two unknowns?
 


You don't need to find where they intersect.

With xy= 1, xy'+ y= 0 so y'= -y/x. With x^2- y^2= 1, 2x- 2yy'= 0 so y'= x/y. Without worrying about what the "points of intersection" are, what is the product of thos two derivatives for a given (x, y)?
 


Thank you!
 

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