Let $C$ be a circle $x^{2} + y^{2} = 1$. The line $l$ intersects $C$ at the point $( - 1,0)$ and the

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Let $C$ be a circle $x^{2} + y^{2} = 1$. The line $l$ intersects $C$ at the point $( - 1,0)$ and the point $P$. Suppose that the slope of the line $l$ is a rational number $m$. Number of choices for $m$ for which both the coordinates of $P$ are rational, is:

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[Key Concept: A line meeting the unit circle at $( - 1,0)$ and another point: parametrise the second intersection.]$m = \frac{sin	heta}{cos	heta + 1} = tan\left( \frac{	heta}{2} ight)$$$P(x,y) = \left( \frac{1 - 	an^{2}\frac{	heta}{2}}{1 + 	an^{2}\frac{	heta}{2}},\frac{2tan\frac{	heta}{2}}{1 + 	an^{2}\frac{	heta}{2}} ight)$$$	herefore\ x,y \in Q$, if $m \in Q \Rightarrow$ Infinite points are possible.

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