Equation of Ellipse

Algebra Level 5

There is an ellipse with focal points

P=(4,2)P = (4, 2)

and

Q=(2,4)Q = (2, 4)

One of the points on the ellipse is (2,2)(2, 2). If the equation for this ellipse is

ax2+bxy+cy2+dx+ey+f=0ax^2 + bxy + cy^2 + dx + ey + f = 0

where gcd(a,b,c,d,e,f)=1\gcd(a, b, c, d, e, f) = 1 find the maximum value of the following expression:

a+b+c+d+e+fa + b + c + d + e + f

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