Conversion of Algebra To Geometry is an art

Geometry Level 5

Let $a,b \in R$ such that $E$ is minimum where $E=\sqrt { { a }^{ 2 }+{ b }^{ 2 } } +\sqrt { { (a-12) }^{ 2 }+{ b }^{ 2 } } +\sqrt { { a }^{ 2 }+{ (b-5) }^{ 2 } }$ Let ${ E }_{ \text{min} }^{ 2 }=m+n\sqrt { l }$ where $m,n$ are co-prime integers and $l$ is a square free integer. Then, find the value of $m+n+l$ .

Hint:

Don't Use Calculus since here it is Poisonous for you.

This is Part of set Click here .

The answer is 232.

2 solutions

Deepanshu Gupta
Sep 13, 2014

$I'am\quad waiting\quad for\quad others\quad to\quad post\quad a\quad full\quad solution\quad for\\ this\quad cool\quad problem.\\ I\quad can\quad give\quad you\quad hint\quad that\quad assume\quad an\quad right\quad angled\\ Triangle\quad ABC\quad in\quad X-Y\quad plane\quad and\quad let\quad A(0,5)\\ \& \quad B(12,0)\quad \& \quad C(0,0).\\ Now\quad let\quad a\quad point\quad P(a,b)\\ Thus\quad we\quad have\quad to\quad calculate\quad minimum\\ value\quad of\quad E=\quad PA+PB+PC\\ By\quad Logically\quad thinking\quad we\quad conclude\quad that\\ 'P'\quad must\quad lie\quad inside\quad the\quad triangle\quad and\quad make\quad equal\quad \\ Angles\quad with\quad respective\quad sides\quad i.e\quad angle=120\quad degree\\ Now\quad use\quad cosine\quad law\quad in\quad respective\quad 3\quad triangles.And\quad Do\quad \\ Some\quad mathamatical\quad Calculation\quad and\quad compute\quad the\quad value\quad of\\ { E }_{ min }^{ 2 }=\quad 169\quad +\quad 60\sqrt { 3 } \\ l=169\\ m=60\\ n=3\\ \Longrightarrow l+m+n=232$ .

I think you meant B(12,0). right???

Chirag Singapore - 6 years, 3 months ago

can you explain how will p make 120 degrees .Actually , I am unable to make the figure.

Raven Herd - 6 years, 6 months ago

It's because in an acute angled triangle the $Fermat's Point$ lies on the pint that makes an angle of $120^o$ with the lines joining the vertices of the triangle to it. For more info. you can google ,"Fermat's Point"

Kunal Gupta - 6 years, 3 months ago

Could you elaborate your calculations? I seem to miss a factor somewhere

Suhas Sheikh - 3 years ago

Does that point lies on the centroid of this triangle for which the value would be minimum

Avi Ghanshani - 5 years, 10 months ago

No the centroid is the point that minimises the sum of the square of the distances.

Spandan Senapati - 4 years, 1 month ago

Nguyễn Phát
Sep 29, 2014

With D(-(5 square root of(3)/2, 5/2)
Using theory of 'Torricelli/Fermat Point Triangle' we have Emin=BD then Emin^2= BD^2 =....
Actually, B(12,0)

Conversion of Algebra To Geometry is an art

Hint:

This is Part of set Click here .

The answer is 232.

2 solutions

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