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 ??

#Geometry #HelpMe! #MathProblem #Math

Easy Math Editor

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Comments

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

Piyushkumar Palan - 7 years, 6 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!!!

Tanya Gupta - 7 years, 3 months ago

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

Devkant Chouhan - 4 years, 7 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!

Finn Hulse - 7 years, 3 months ago

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

Sai Kalyaan Palla!!! - 3 years, 2 months ago

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 - 7 years, 6 months ago

cool

jafar badour - 6 years, 3 months 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

Saswata Dasgupta - 6 years, 1 month ago

lagrange methode in calculus maybe help

Abdul Siregar - 7 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$