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Minimum Distance Between 2 Curves

I was trying ( unsuccessfully ) to find the minimum distance between two parabolas and thought that I could do that by finding the minimum distance between two parallel tangents to the two parabolas. However there comes a case when ( as shown in figure ) the tangents are indeed parallel and the distance between them is also minimum but ( as shown by the green line ) the actual distance between the point of contacts is not the distance between the parallel lines but much more ... How do I do such problems ??

Note by Santanu Banerjee
4 years ago

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12 votes

  Easy Math Editor

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Minimum distance will be along common normal. Use little calculus & co-ordinate geometry to get it.

Piyushkumar Palan - 3 years, 12 months ago

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I don't think calculus will be needed....work using the parabola y^2=4ax....and use the parametric form of the equation i.e. (at^2,2at)...then write equation for both the normals and equate them....works beautifully!!!

Tanya Gupta - 3 years, 8 months ago

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How can we equate them there would be two variable t1 and t2

Devkant Chouhan - 1 year ago

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You can find the symmetry line.......as an example y=x is the line of symmetry between y^2=4x and x^2=4y double the dist from one parabola to y=x and you get the distance

Saswata Dasgupta - 2 years, 6 months ago

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Ooh! My favorite! You have to set up a distance formula, with each equation as a point! It's awesome! From there, you simplify, and use basic algebra to minimize! Great post!

Finn Hulse - 3 years, 8 months ago

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cool

Jafar Badour - 2 years, 8 months ago

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can be solved by making the equation in a variable involving parametric equations for the two curves .. try to get it in one single parameter and then differentiate to get the critical point.. !!

Ramesh Goenka - 3 years, 12 months ago

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lagrange methode in calculus maybe help

Abdul Siregar - 3 years, 12 months ago

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