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Does there exists two different curves from $ \ (0,0) \ \ to \ \ (10,0) \ $ having same arc length ?

Answer:

I think there does not exists such two different curves.

For, one of the curve is the straight line segment $ \ x=10t , \ 0 \leq t \leq 1 \ $

This line segment has curve length $ \ =10 \ $

Any other curve joining $ (0,0) \ \ to \ \ (10,0) \ $ must be different with different arc length.

Help me with better way.

MAS
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3 Answers3

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Well yes, there are: Take an arbitrary curve $l$ which is not a straight line. Then take the straight line between the points and reflect $l$. This way you get a second curve having the same arc length by symmetry.

asdf
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  • Any continuous function $y=f(x)$ that has roots $(0, 0)$ and $(10, 0)$ will have another function $y=-f(x)$ that has the same roots and equivalent arc lengths between the two points. – dan post Mar 29 '18 at 07:37
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Guide:

Try to construct a circle with centered at $(5,0)$, making it passes through $(0,0)$, and $(10,0)$.

Siong Thye Goh
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    It joins $ \ (0,0) \ \ and \ \ (10,0 ) \ $ not $ \ (10,10) \ $ – MAS Mar 29 '18 at 06:46
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    @yourmath Then a circle centered ad $(5,0)$. How about you try and focus on the point of the answer, not on the tiny mistake made in the answer? – 5xum Mar 29 '18 at 06:48
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    thanks for pointing out the mistake and the quick fix guys. appreciate it. – Siong Thye Goh Mar 29 '18 at 06:51
  • the curve is $ x(t)=\sqrt 5 \cos t , \ y(t)=\sqrt 5 \sin t , \ 0 \leq t \leq \pi \ $ . The arc length is $ =\sqrt 5 \pi \ $ . which is the other curve? – MAS Mar 29 '18 at 07:00
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    hmm... I don't think you have found the right description, try to see what do you get when you substitute $t=0$ into your equation. Don't give up, give it another try. The whole idea is that it is a circle, if you have found the curve that is upper semicircle, recall that it is symmetical about the $x$-axis, you just have to reflect it, the second curce can be obtained by multipling $-1$ to the $y$ part. – Siong Thye Goh Mar 29 '18 at 07:11
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You have asserted that there are not two different curves with the same arc length, and your proof is that no curve has the same length as one particular curve you have given. This is not a proof, any more than "15 is prime, because 2 doesn't divide it" is.

Patrick Stevens
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