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

parabola is a locus of a point whose dist from a fixed point and fixed line are equal.

[(3x+7y-21)^2]/(58)^1/2=[(x-3)^2+(y-3)^2]^(1/2)

solving we get

49x^2+9y^2-42xy-222x-54y+603=0

gcd=1

answer=49+9-42-222-54+603 =343

Given the focus is at F(3,3) and let P(x,y) be a point on the parabola.

By definition of parabola, the distance between the focus (F) and a point on the parabola is equal to the distance bewteen the point of the parabola P and the line 3x+7y=21, i.e.,

Sqrt [ (x-3)^2+(y-3)^2) ] = | 3x+7y-21|\ sqrt(3^2+7^2)

Since we are just solving for the sum of the coefficients of the terms once expanded and simplified theb we just let x=y=1. So, we have

Sqrt [ 4+4] = |3+7-21|\sqrt(58)

Sqrt [8(58)]= |-11|

Square both sides,

343-121=343.

So the answer is 343.

@Andres Rico M. III Gonzales

LOL maximum has nothing to do with that expression answer is simply 343

Let $\displaystyle \left( h,k \right)$ be any point on the parabola. Then by the definition of the parabola, it must satisfy the following equation: $\displaystyle \sqrt { { \left( h-3 \right) }^{ 2 }{ +\left( k-3 \right) }^{ 2 } } =\cfrac { 21-3h-7k }{ \sqrt { { \left( 3 \right) }^{ 2 }{ +\left( 7 \right) }^{ 2 } } }$ .

On simplifying the equation and bringing it down to the form $\displaystyle a{ x }^{ 2 }+bxy+c{ y }^{ 2 }+dx+ey+f=0$ , and then comparing the coefficients then adding $\displaystyle \left[ i.e.\quad a+b+c+d+e+f \right]$ we get the answer as $\displaystyle \boxed{343}$ .

Here is a glance of the computation $\sqrt { { \left( h-3 \right) }^{ 2 }{ +\left( k-3 \right) }^{ 2 } } =\cfrac { 21-3h-7k }{ \sqrt { { \left( 3 \right) }^{ 2 }{ +\left( 7 \right) }^{ 2 } } } \\ \Rightarrow { \left( h-3 \right) }^{ 2 }{ +\left( k-3 \right) }^{ 2 }=\cfrac { { \left( 21-3h-7k \right) }^{ 2 } }{ 58 } \\ \Rightarrow { h }^{ 2 }-6h+9+{ k }^{ 2 }-6k+9=\cfrac { { \left( 21-\left( 3h+7k \right) \right) }^{ 2 } }{ 58 } \\ \Rightarrow 58\left( { h }^{ 2 }-6h+9+{ k }^{ 2 }-6k+9 \right) ={ 21 }^{ 2 }+{ 9h }^{ 2 }+{ 49k }^{ 2 }+42hk-42.3h-42.7k\\ \Rightarrow 58{ h }^{ 2 }+58{ k }^{ 2 }-348h-348k+1044=441+9{ h }^{ 2 }+{ 49k }^{ 2 }+42hk-126h-294k\\ \Rightarrow 49{ h }^{ 2 }+9{ k }^{ 2 }-22h-54k-42h+603=0$ .