An easy way to do this is to parameterize the directed line segment from B to A by the function
[tex]\ell_{B\to A}(t) = (1-t) B + t A = (1 - t) (11, 17) + t (-9, 4)[/tex]
with 0 ≤ t ≤ 1.
The point P₂ splits up AB so that BP₂ = 3/8 AB and AP₂ = 5/8 AB. Then we reach the point P₂ when t = 3/8, so its coordinates are
[tex]P_2 = \ell_{B\to A}}\left(\dfrac38\right) = \dfrac58 (11,17) + \dfrac38 (-9,4) = \boxed{\left(\dfrac72, \dfrac{97}8\right)}[/tex]
