I was reading Rational Points On Elliptic Curves and after the first chapter, this is a question I found-
Describe all rational points on the circle $$x^2 + y^2 = 2$$ by projecting from the point $(1, 1)$ onto an appropriate rational line (Your formulas will be simpler if you are clever in your choice of the line).
Now, I know how to do this projection and find out the rational points. However, I was stuck at
Your formulas will be simpler if you are clever in your choice of the line
My question is, is there any way to understand beforehand without doing any calculations (or doing minimum calculations) which line is the most suitable one to perform this projection?
Also, the question specifies the point $(1,1)$. What if it was not specified? Do we have any way to make a guess for the easiest choice of this point?
And, I am not just talking about this circle. Is there any general way to guess the best choice for a general circle $x^2+y^2=r^2$?
Please note that I am aware of some other methods that can lead us to the solution of this problem. But, I am specifically asking for the clever choice of the line and the point for a general circle. In other words, is there any kind of a hack to know from beforehand whether one choice of a point and a line is better than the other?