Question

find the ratio in which the line y = 4- 2x , divides the line -segment joining the points a (2,-2) and b(3,7)

Answer 1

L1:   y = 4 - 2 x        -- (1)

L2 :  AB   where A = (2, -2)   , B = (3,7)
        Slope of AB = (7+2) / (3-2) = 9
        equation:   (y - 7) / (x - 3) = 9
                          or    y = 9 x - 20    --- (2)

Let C (x1, y1) be the point of intersection between L1 & L2.
         so   y1 = 4 - 2 x1 = 9 x1 - 20
                x1 = 24/11
                y1  =  -4 /11

Let  C divide  AB in ratio   m : (1-m)  ,         m is a fraction.
       x1 = 24/11= 3 m + 2(1-m) = m +2
           => m = 2/11
       y1 = -4/11 = 7m -2 (1-m) = 9 m - 2    => m = 2/11     

Ratio:     2/11 :  9/11       or   2 : 9