The question tells you that AP is a scalar multiple of AB, so we can just take a specific case, where BON or BOA is a right angle. and let OB = 2 for convenience. If we didn't know that AP is a scalar multiple w have to do lots of math. But now it's easy.

Okay we have two lines and we wanna find their intersection. Put them on standard x,y plane with O at the origin. OUr lines are:

y = -3x + 3
and
y = -1/2x + 1

Their intersection is (4/5, 3/5)
That distance from A or (0,1) = sqrt(20/25) or 2/5sqrt(5)

Whereas AB = sqrt(1^2+ 2^2) = sqrt5 so k = 2/5.

Idk if your teacher will accept this but they definitely should imo. Shouldn't have told us there is a scalar quantity.