Tangent of two curves

  • Thread starter songoku
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  • #1
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Homework Statement


let C1 : y = x - 1/2 x2 and C2 : x = y - 1/2 y2 be curves on the xy plane.

1. find the equation of the tangent to the curve C1 at x = k

2. suppose the line obtained in 1) is also tangent to the curve C2. find all values of k and the equations of the tangents.

3. evaluate the area of the figure enclosed by all tangents obtained in 2) and the curve C2

Homework Equations


derivatives, equation of tangent


The Attempt at a Solution


1. I've done it. I got : y = (1-k) x + 1/2 k2

2.
differentiate C2 with respect to x :
1 = dy/dx - y dy/dx

dy/dx = 1/(1-y) = m

so, 1/(1-y) = 1-k

then.....I...gave up...
 

Answers and Replies

  • #2
rock.freak667
Homework Helper
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If x=k is a point on C1, then the point (k,k-1/2k2) ((x,y)) is a point on the line right?


So the gradient of the tangent on C2 is dy/dx= 1/(1-y). Since (k,k-1/2k2) lies on the line, what is the gradient of this tangent?


then put that equal to 1-k.
 
  • #3
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If x=k is a point on C1, then the point (k,k-1/2k2) ((x,y)) is a point on the line right?


So the gradient of the tangent on C2 is dy/dx= 1/(1-y). Since (k,k-1/2k2) lies on the line, what is the gradient of this tangent?


then put that equal to 1-k.

Do you mean substituting y = k - 1/2 k2 to dy/dx= 1/(1-y) ?
 
  • #4
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sorry (editted)

alternative way,

you also can find another tangent equation with gradient 1/(1-y) with point (k,k-1/2k2), and compare with the other tangent eqution
 
  • #5
1,837
181
sorry (editted)

alternative way,

you also can find another tangent equation with gradient 1/(1-y) with point (k,k-1/2k2), and compare with the other tangent eqution

I don't get it. x = k is the common point between the tangent and C1 and I think we can't use it to find the equation of tangent of C2 since x = k may not be the common point.

Or maybe I missed the hint?
 
  • #6
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it says that, the tangent of C1 at x=k , is also tangent C2 at some point x, implies that the tangent of C1 and tangent of C2 is the same line.

since it is the same line, tangent C2 also pass through point (k,k-1/2k2)
 
Last edited:
  • #7
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i'm sorry but i think i'm wrong
 
Last edited:

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