It is easy to make the straight line AB y=kx+b, because it is brought in through P, and the relationship between K and B is b= root number 3-9k, so y=kx+ (root number 3-9k).

(The origin is O) Let x=0 and y=0 respectively to get the length of OA and OB. According to Pythagoras, AB square =OA square +OB square,

AB= root number (root number 3-9k) square+(root number 3-9k)/k square, this is a function about k, so I won't talk about it below! Find the maximum value of the function (but I don't know what grade you are in, have you ever studied, as long as you have knowledge in the first grade)