Conversion of Algebra To Geometry is an art

Geometry Level 5

Let a,bRa,b \in R such that EE is minimum where E=a2+b2+(a12)2+b2+a2+(b5)2E=\sqrt { { a }^{ 2 }+{ b }^{ 2 } } +\sqrt { { (a-12) }^{ 2 }+{ b }^{ 2 } } +\sqrt { { a }^{ 2 }+{ (b-5) }^{ 2 } } Let Emin2=m+nl{ E }_{ \text{min} }^{ 2 }=m+n\sqrt { l } where m,nm,n are co-prime integers and ll is a square free integer. Then, find the value of m+n+lm+n+l.

Hint:

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

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