When is $\frac{a\sqrt{2}+b}{b\sqrt{2}+c}$ an integer?

Question

$\frac{a\sqrt{2} + b}{b\sqrt{2}+c}$ is a number, where $a, b, c$ are integers. What should be the condition for above number to be an integer? One possible solution is $a = b = c$. Other solutions would be a great help.

Answer 1

$$ \frac{a\sqrt{2}+b}{b\sqrt{2}+c} = t \iff a\sqrt{2}+b=bt\sqrt{2}+ct \iff a=bt, \ b=ct \iff a=ct^2, \ b=ct $$ because $\sqrt{2}$ is irrational.