How do I find all positive integer pairs $(a, b)$ of a hyperbola of form $\frac{ab-2a}{b}=4$?
The standard way of doing that, factorisation, doesn't seem to work here.
I did try solving for $b$ and inserting that in the equation, but that doesn't give me any other information other than $a > 4$.
These equations don't seem to lead me to a solution, either.
- $b=\frac{2a}{a-4}$
- $ab-2a=4b$
- $4b+2a=ab$
Any help?
$4+2=$ can be written as $(a-4)(b-2)=8$. Hence there are 4 possibilities:
$a-4=1,\ b-2=8$, i.e. $(a,b)=(5,10)$.
$a-4=2,\ b-2=4$, i.e. $(a,b)=(6,6)$.
$a-4=4,\ b-2=2$, i.e. $(a,b)=(8,4)$.
$a-4=8,\ b-2=1$, i.e. $(a,b)=(12,3)$.
These are all solutions where $a-4>0$, $b-2>0$. However, there are more positive solutions when they are negative, but none of them will satisfy the condition $a,b>0$. I list them here just in case anyway:
$a-4=-1,\ b-2=-8$, i.e. $(a,b)=(3,-6)$.
$a-4=-2,\ b-2=-4$, i.e. $(a,b)=(2,-2)$.
$a-4=-4,\ b-2=-2$, i.e. $(a,b)=(0,0)$.
$a-4=-8,\ b-2=-1$, i.e. $(a,b)=(-4,1)$.