My problem is that I can't create a line that passes through both points AND is the shortest path AND touches both axis lines.
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Hint: A path can be a sequence of segments. You can go from $(8,2)$ to $(0,y)$ to $(x,0)$ to $(3,5)$ Calculate the length of each segment, add them to get the total path length. Then you can differentiate to find the minimum in each variable-you will get two equations in two unknowns. Or, you can remember angle of incidence=angle of reflection and use that. I don't warrant that this you should go to the $y$ axis first, but it looks likely.
Hint: fix one point and look at paths from this point to a reflection of the other in both axes (you need to cross both) - eg from (3,5) to (-8,-2)
Think about how the proof by reflection works for one mirror, and apply it with two.
You follow path of light.
First point is (3,5), then comes (0,a) and (b,0) and finally (8,2).
Because it's reflection in both cases, $k_1=-k_2$ and $k_2=-k_3$: