Can we find $x$ and $y$ positive integers such that $f(x,y)$ is a strictely positive integer Let us consider the function:
$$f(x,y)=\dfrac{axy+by+cx+d}{rxy+sy+hx+w}$$
Here $f(x,y)$ is the division of two quadratic polynomials with the same degree
where $a,b,c,d,r,,s,h,w$ are non zero integers.
My question is:
(1) Given $a,b,c,d,r,,s,h,w$ as integers, can we find $x$ and $y$ rational numbers such that $f(x,y)$ is a strictely positive integer.
(2) Given $a,b,c,d,r,,s,h,w$ as integers, can we find $x$ and $y$ positive integers such that $f(x,y)$ is a strictely positive integer.
add
We can assume that $f$ is bijective with respect to $x$ if $y$ is a constant and bijective with respect to $y$ if $x$ is a constant.
 A: The denominator can be the opposite of the numerator, so that the function is a negative constant. Take for instance $$\frac{xy+x+y+1}{-xy-x-y-1}$$
 in this way $f(x, y) = - 1$ and you have no chance of finding such $(x, y) $! 
Let me enounce a simple fact:
For every fixed integers $u, v$, it exists an integer N of the same sign of $u$ such that for every $n$ of the same sign of $u$ with $|n|>|N|$ we have
$$un+v >0 $$
As a corollary, we have :
If u, u' have the same sign, for every v, v' it exists n such that un+v, u'n+v' are positive.
Proof. Take N, N' given by the previous fact: we know they have the same sign. Thus if we take a n of the same sign of N, N', but greater in modulus, we have $un+v, u'n+v'$ both positive. 
Now i will show you that if $a, r$ have the same sign, the answer is affirmative in both cases. In fact, choose $x$ of the same sign of $a$, big enough such that both $ax+b, rx+s$ are positive. Then you can find $y$ such that both the linear forms $(ax+b) y + (cx+d), (rx+s) y+(hx+w) $ are positive, because the coefficient of $y$ have the same sign. In conclusion, the quotient
$$\frac{axy+by+cx+d}{rxy+sy+hx+w} $$
Is positive! 
