Indeterminate equations

To turn the tables..............We can also solve this using Co-ordinate Geometry....... Thanks to @Rishvic Pushpakaran !!!

Nice question.

By using linear diophantine equations

We can easily determine that

$x= 4c+9t$ & $y=-3c-9t$ .

$\forall t \in N$

Now since both $x$ and $y$ are positive

We find $-3c/7 >t > -4c/9$ .

Now for maximum put $c= 63\kappa$ .

$\forall \kappa \in N$ .

Thus $-27\kappa > t > -28\kappa$ .

Now to have 6 values of $t$ , $\kappa=7$ .

Thus answer=63*7= $\boxed{441}$ .