Help please =?!!

I recently gave a sectional test and came across this question the second time. I didn't remember the way to solve this question though. I thought, "Oh, I might get an answer on the straight line connecting both the points", time was running fast and at the last moment i did not understand anything and thought that "maybe keeping the point A in the middle of the connecting line and making it equidistant from both the points may result in least distance from both the extreme points. I want to know if I have applied the correct logic. Also please tell why the solution uses this approach because I'm unable to understand how!!! can we randomly say anything like a ellipse will answer, then take equal x and y points and stuff like this.

Please help! Genuinely perplexed.