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
5 years, 5 months ago

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

- 5 years, 5 months ago

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!!!

- 5 years, 1 month ago

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

- 2 years, 5 months ago

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!

- 5 years, 1 month ago

so its like distance between (x,f(x)) and (y,g(y))

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.. !!

- 5 years, 5 months ago

cool

- 4 years, 1 month ago

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

- 3 years, 11 months ago

lagrange methode in calculus maybe help

- 5 years, 5 months ago