3. Let a circle and a line be given.

a) Construct a circle with a given radius that touches the given circle and the given line.

b) Discuss the number of solutions to the given problem.
Solution. a) Let $p$ be the given line, $k(O, r)$ the given circle, and $k(S, R)$ the required circle (see diagram). Since $k(S, R)$ touches $p$, it follows that $S$ lies on the line $a$, parallel to $p$ and at a distance $R$ from it. Similarly, $S$ must

![](https://cdn.mathpix.com/cropped/2024_06_05_81dca0670dc808162b1cg-3.jpg?height=289&width=510&top_left_y=1480&top_left_x=932)
lie on the circle $k(O, R+r)$ or $k(O,|R-r|)$, so $S$ will be the intersection of that circle with the line $a$.

b) The problem can have at most 8 solutions, because two lines and two circles can intersect at most in 8 points, and there are at least two solutions.

The problem will have no solution if the line $p$ and the circle $k(O, R+r)$ or $k(O,|R-r|)$ do not intersect, which is possible when $2R + r$ is less than the distance from $O$ to the line $p$.