4. The rod $AB$ of a thermometer hanging vertically on a wall has a length of $2r$. The eye of the observer is on a line $l$ which is normal to the plane of the wall and intersects the line $AB$ at a point whose distance from the midpoint of the segment $AB$ is equal to $h(h>r)$. At what distance from the wall should the observer's eye be located so that the angle under which the observer sees the rod is the largest?
Solution. Let $k$ be the circle that contains points $A$ and $B$ and point $C \in l$, and let $O$ be the center of this circle (diagram on the right). From point $C$, the observer sees the segment $A B$ at an angle $\angle B C A = \frac{1}{2} \measuredangle B O A$. This angle is the largest when the distance from point $O$ (the center of the circle that has common points with line $l$) to line $A B$ is as small as possible. It is easily proven that this is achieved if $k$ is tangent to line $l$, and in this case, the distance from point $C$ to line $A B$ is

![](https://cdn.mathpix.com/cropped/2024_06_05_45d0daa1fd152b0afea5g-6.jpg?height=526&width=507&top_left_y=631&top_left_x=972)
equal to $\sqrt{h^{2}-r^{2}}$, and this is the desired distance.

## 4th Year