An east west and north south road intersect at a point O. A diagonal road is to be constructed from a point A east of O to a point B north of O, passing through a town C which is $a$ miles east and $b$ miles north of O. Find the ratio of OA to OB if the triangular are OAB is as small as possible.Show that this minimal area is attained when C bisects the segment AB.
My approach is $\frac{B-b}{a}=\frac{b}{A-a}$ And $\frac{AB}{2}=Area_{min}$ so what is $\frac{A}{B}$ then it doesnt flow from here. I couldnt figure out which is fixed. Like everthing depent on each other.Do you have opinion?
Your approach is a good start. Now solve the first equation for $A$ and plug it into the area equation. That gives you the area as a function of $B$. Take the derivative, set to zero...