I was helping a somebody on a problem when a wild question appears. It goes like this:
A fly stationed at a point on the circumference of the base of a cylindrical tower of radius $12$ feet. Find that he can just see a distant flagstaff by walking along the tangent line to the base either a distance $8$ feet in one direction or $5$ feet in the other direction. Find the distance of the foot of the flagstaff from the center of the base of the tower.
My work
This is what I imagined...
Adding some finer details...
We see the figure above that...
$$x^2 = 4^2 + z^2$$ $$y^2 = 7^2 + z^2$$
Then...
$$x^2 = 4^2 + (y^2 - 7^2)$$ $$x^2 = y^2 - 33$$ $$x^2 - y^2 = -33$$
I guess I'm stuck. We got one equation and two unknowns. How to answer the above question?
The following figure is a view of the tower from above.
The shaded disk is the base of the tower. Point $F$ is the starting point of the fly on the circumference of the base of the tower. The fly is constrained to walk along a line tangent to the base of the tower; the only such line that includes the fly's starting position is the line $GFH.$ The points $G$ and $H$ are respectively $8$ feet from the fly's starting position in one direction along the tangent line and $5$ feet from the fly's starting position in the other direction along the tangent line.
Region $A$ (including line $GFH$ and all points to its left) contains points all of which are visible from $F,$ $G,$ $H,$ and any point between those points; no point in that region is "just" visible from either $G$ or $H.$ Points in regions $B$ and $C$ are not visible from $H$; points in regions $C$ and $D$ are not visible from $G.$ No point is "just" visible from $G$ and "just" visible from $H,$ so there is no point at which a flagpole can be placed that satisfies the conditions in the problem statement.
Conclusion: The statement of this problem is a mistake. It has no solution.