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Homework Help Overview

The discussion revolves around finding the equation of the tangent to the curve defined by xy=4 at a specific point P, as well as proving certain properties related to points A and B where the tangent intersects the axes. The problem also involves finding normals to the curve that are parallel to a given line.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the equation of the tangent line and its implications for points A and B. There are attempts to find coordinates for these points and to prove that P is the midpoint of AB. Questions arise about how to calculate the area of triangle OAB and the conditions under which it remains constant. Some participants express uncertainty about the calculations needed for the normals.

Discussion Status

There is ongoing exploration of the relationships between points A, B, and P, with some participants confirming findings and others questioning the correctness of their equations. Guidance is offered regarding the slopes of the normals and their relationship to the tangent line, but no consensus is reached on the final calculations for the normals.

Contextual Notes

Participants note potential errors in the equations derived for the tangent line and the normals, as well as the need for specific points to complete the calculations. There is an acknowledgment of the assumptions made regarding the positivity of certain variables.

lionely
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Find the equation of the tangent to the curve xy=4 at the point P whose coordinates are (2t,2/t). If O is the origin and the tangent at P meets the x-axis at A and the y-axis at B, prove
(a) that P is the midpoint of AB
(b) that the area of Triangle OAB is the same for all positions of P.
Find the the equations of the normals to the curve xy=4 which are parallel to the line
4x-y-2=0.


I got the equation of the of the tangent at P to be yt2 - 4t -x = 0

Not sure how to do a and b I don't really have much information about AB other than it's points like on P basically.
 
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The y-coordinate of A is 0. It lies on the tangent, so it satisfies yt2 - 4t -x = 0. What is its x-coordinate?
 
x is -4t
 
lionely said:
Find the equation of the tangent to the curve xy=4 at the point P whose coordinates are (2t,2/t). If O is the origin and the tangent at P meets the x-axis at A and the y-axis at B, prove
(a) that P is the midpoint of AB
(b) that the area of Triangle OAB is the same for all positions of P.
Find the the equations of the normals to the curve xy=4 which are parallel to the line
4x-y-2=0.

I got the equation of the of the tangent at P to be yt2 - 4t -x = 0

Not sure how to do a and b I don't really have much information about AB other than it's points like on P basically.
Now that you have the coordinates for point A, find the coordinates for point B. (The x coordinate for point B is 0.)
 
B is (0, 4/t) oh and when you work it out the mid point is in fact P... to do part b now do I find the length of the lines and work out the area?
 
lionely said:
B is (0, 4/t) oh and when you work it out the mid point is in fact P... to do part b now do I find the length of the lines and work out the area?
Draw a sketch .

The result should be pretty obvious.
 
34fdwsm.png


So no calculations are needed to prove it?

How would I do the final part if I don't know any points for the normals?
 
lionely said:
34fdwsm.png


So no calculations are needed to prove it?

How would I do the final part if I don't know any points for the normals?
I just noticed that must be an error in your equation for the tangent line. Assuming that t is positive, both intercepts should be positive.

You have the wrong sign on x.
 
This should be okay now right?

dxicxw.png
 
  • #10
Area = 1/2 * 4t * 4/t = 8

Hence constant.
 
  • #11
lionely said:
I got the equation of the of the tangent at P to be yt2 - 4t -x = 0


yt2 - 4t + x = 0
 
  • #12
Yeah now I see it, thanks. But the final part of the question
Find the the equations of the normals to the curve xy=4 which are parallel to the line
4x-y-2=0.

How would I do this without having any points for the normals?
 
  • #13
I get the normal as y=4x-15
 
  • #14
How did you calculate it I don't understand we don't have any points.

:S
 
  • #15
lionely said:
How did you calculate it I don't understand we don't have any points.

:S

I'm not sure I am right.

But, Slope * Slope_normal = -1

You know Slope of curve so find slope of normal.

Then equate it to slope of given line to find where the normal lies.
 
  • #16
slope of curve = -4x^2 = equation of given line 4x-2

solve for x? then get y from equation of the curve then find equation of normal?

IGNORE this makes no sense.
 
Last edited:

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