Another Turning point question

In summary, the conversation discusses finding the equation of the tangent to the curve xy=4 at the point P(2t,2/t), proving that P is the midpoint of AB and that the area of Triangle OAB is the same for all positions of P. It also discusses finding the equations of the normals to the curve xy=4 which are parallel to the line 4x-y-2=0. The equation for the tangent at P is yt2 - 4t -x = 0. To prove that P is the midpoint of AB, the coordinates for point A are found to be (-4t, 0) and the coordinates for point B are (0, 4/t). The final part of the
  • #1
lionely
576
2
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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  • #2
The y-coordinate of A is 0. It lies on the tangent, so it satisfies yt2 - 4t -x = 0. What is its x-coordinate?
 
  • #3
x is -4t
 
  • #4
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.)
 
  • #5
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?
 
  • #6
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.
 
  • #7
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?
 
  • #8
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.
 
  • #9
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:

What is "Another Turning Point"?

"Another Turning Point" is a phrase that is often used to describe a significant event or moment in time that results in a change of direction or perspective.

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Some examples of "Another Turning Point" in scientific history include the discovery of DNA structure, the theory of evolution by natural selection, and the development of the periodic table.

Can "Another Turning Point" be predicted or planned?

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