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.

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