We have a directrix and we have a focal point. What's missing?

Algebra Level 5

There is a parabola with directrix defined by the line

3x+7y=213x + 7y = 21

and focal point

P=(3,3).P = (3, 3).

The equation of the parabola can be written as

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

where the gcd(a,b,c,d,e,f)=1\gcd\big(|a|, |b|, |c|, |d|, |e|, |f|\big) = 1. Find the maximum value of

a+b+c+d+e+f.a + b + c + d + e + f.

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