A correct solution came from Bradley student
Andrew Greene.
Other correct solutions were submitted by
Farid Lian, Bill Webb, Juan Carlos Marivela, Lou Cairoli, Cee Ann Franklin
and Colin Shuker.
The answer to the general problem is that mn=b2/a2. Applied to the specific case, this means that m=1/12.
If the ellipse were a circle, it would be clear by symmetry that the slopes should be reciprocals. Luckily, we can move our problem to this situation by changing variables. Set x:=au and y:=bv. These stretches don't change the ratio between the areas. They turn the x,y-ellipse into a u,v-circle and take the lines y=mx and y=nx to the lines v=nau/b and v=mau/b. Making these u,v-lines have reciprocal slopes gives the result.