We have a unit circle $C:x^2+y^2=1$. Let $l:y=m(x+1)$.
We consider a circle $C'$ at a centre on $l$ that is inscribed to an upper semi-circle, i.e., a circle that is tangent to the circle $C$ and the $x$-axis.
How can I describe the centre of $C'$ as a function of $m$, the slope of $l$?
What is the locus of the centers of circles inscribed to an upper semi-circle?
If you let $P$ be such a center, then the ray of the circle $C'$ is equal to its $y$ coordinate $P_y$, ad if you draw the line $OP$, with $O$ the origin $(0,0)$, it meets $C$ in a point $Q$, but $C'$ is tangent to $C$, so $PQ$ must again be the ray of $C'$, and applying Pitagora theorem on the triangle $O -- (P_x,0) -- P$ you obtain $$P_x ^ 2 + P_y^2 = (1-P_y)^2$$ so the locus of the centers is $x^2 +2y -1 = 0$.
The intersection with the line $l$ gives you the point you're looking for.
