The coordinates of the 3 vertices of the triangle

are

\(\left( -2,-2 \right) ,\left( 6,13 \right) ,\left( 10,3 \right) \)

The implicit equation of the inscribed ellipse with the maximum possible area can be expressed in this form

\(a{ y }^{ 2 }+bxy+{ cx }^{ 2 }+dy+ex+f=0\)

where \(a, b, c, d, e, f\) are co-prime integers, and \(a>0\).

Find \(\sqrt { (a+b+c+d+e+f) } \)

You're welcome to check Calvin Lin's post

or try something else.

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