Consider coordinates $B=(-16,0)$ and $A=(12,8).$ Let $D=(x,y)$ be a coordinate of the ellipse $9x^2+25y^2=3600$ such that the distance of $AD+DB$ is minimized. For what value of $(x,y)$ is $AD+DB$ minimzed and what will the minimal distance be?
Firstly, I noticed that $(-16,0)$ is one of the foci of the ellipse (the left one).
After doing some graphing and testing values, I noticed that the distance of $AD+DB$ was minimized when $\angle ADB$ was a right angle. (Does this hold true for all ellipses? If so, why does it?). It seems like I have some information, but I'm not sure how to apply it to solve the problem. If $AD+DB$ is minimzed in this configuration, I would still need the coordinate of $D$ to find $AD$ and also $\sqrt{BA^2 + AD^2}=DB.$ However, I'm not fully sure how to do this. Thanks in advance.
Edit: I find that $AD+BD$ is approximately $29.18+1.9,$ which is close to $31.$ However, would there be a mathematical way to do this and get a more precise answer?
Best Answer
There is a simple geometrical solution. Sum of distances is minimum when the ellipse of foci $A$, $B$ passing through $D$ is tangent to the given ellipse. Hence both ellipses must have the same normal at $D$, i.e. angles $BDA$ and $BDC$ must have the same bisector and that implies they must coincide. Hence point $D$ gives a minimum sum $AD+DB$ if points $CAD$ are aligned.