Let $C$ be a circle $x^2 + y^2 = 1$. The line $y = mx + m$ intersects $C$ at the point $P$ other than $(-1, 0)$, the number of rational choices for $m$ for which both the coordinates of $P$ are rational, is:
Step-by-Step Solution
Key Concept: The parametric form represents points on a circle, and rational slopes generate infinitely many rational points on it.
Given $m = \frac{\sin\theta}{\cos\theta + 1} = \tan\left(\frac{\theta}{2}\right)$, we substitute into the parametric equation $P(x,y) = \left(\frac{1-\tan^2\frac{\theta}{2}}{1+\tan^2\frac{\theta}{2}}, \frac{2\tan\frac{\theta}{2}}{1+\tan^2\frac{\theta}{2}}\right)$. This simplifies to a circle equation. Since $m = \tan\left(\frac{\theta}{2}\right)$ can take any real value and each value gives a point on the locus, infinitely many points are possible when $x, y \in \mathbb{Q}$ and $m \in \mathbb{Q}$.
Correct Answer: 4