Find a formula for all of the points $(x,y)$ on the hyperbola $x^2 - 2y^2=1$ whose coordinates are rational numbers.

Question

I use the initial point as $(-1,0)$ ,then $y=m(x+1)$ and put the y in hyperbola, but I can't find a rational value of m.

Answer 1

\begin{align*} 1 &= x^2 - 2y^2 \\ &= x^2 - 2(m(x+1))^2 \\ &= (1-2m^2)x^2 - 4m^2 x-2 m^2 \text{.} \end{align*} If $m = 0$, $x = \pm 1$ (footnote 1), otherwise $x = \frac{-1-2m^2}{-1+2m^2}$ and then $y = \frac{x^2-1}{2} = \frac{\left( \frac{-1-2m^2}{-1+2m^2}\right)^2 - 1}{2} = \frac{4m^2}{(1-2m^2)^2}$.

(footnote 1): We know $x = -1$ gives a rational point, but no choice of $m$ will give this point, so we have to call it out separately.