Find all rational points on the circle $x^2 + y^2 = 2$ by projecting from the point $(1, 1)$ onto an appropriate rational line

Question

This question comes from Rational Points on Elliptic Curves (Silverman & Tate), exercise $1.5$.

I have tried to be clever about the line I chose, but none of the computations seemed easy. I chose the x-axis, the y-axis, $y= -x$ and $x =\sqrt{2}$. The book says that the computations will be easy if I choose the right line.

Answer 1

Hint: let

$$ y-1 = \frac{m}n (x-1) $$

Answer 2

I’ll go further than @GlenO in explaining one method at least. By taking, as he says, $y=t(x-1)+1$, you now express $x^2+y^2-2$ as a polynomial in $x$ only, for which you know that $x-1$ is a factor. Divide, and get something of the form $Ax+B$ for the other factor, where $A$ and $B$ are polynomials in $t$. Then $x=-B/A$, a rational function in $t$. This gives the $x$-coordinate of the other point of intersection of the circle and the line, and you get the $y$-coordinate by plugging back into your linear relation $y=t(x-1)+1$. This method does not yield easy computations, and I don’t know what the author had in mind. But, for instance, $t=-1/2$ gives you the point $(\frac15,\frac75)$, surenough on the circle.