Rectangle Coordinates System

a)
let P be (x,y)
x = (1(-1) + r(5))/(1 + r) = (5r - 1)/(1 + r)
y = (1(3) + r(-8))/(1 + r) = (3 - 8r)/(1 + r)
ans: P is ((5r - 1)/(1 + r) , (3 - 8r)/(1 + r))
b)
assume line AB and the line x - 2y - 5 = 0 intercept at P
Put P into the eqn., we have
(5r - 1)/(1 + r) - 2(3 - 8r)/(1 + r) - 5 = 0
(5r - 1) - 2(3 - 8r) - 5(1 + r) = 0
5r - 1 - 6 + 16r - 5 - 5r = 0
16r = 12
r = 12/16 = 3/4
AP : PB = r : 1 = 3/4 : 1 = 3 : 4