Question

Calculate the coordinates of the point P which divides the line joining A(-1,3), B(5,9) in the ratio 1:2.

Answer 1

See image.......I have solved on paper.....................
Mistake resolved. Verified graphically, answer is 100% right

Answer 2

We can do by three methods.

One: By writing the formula for distance between two points AP(x,y) and PB and equating their ration to 1:2.  Difficult calculations.

Two : By shifting the origin to A(-1,3).  Then A' in new coordinate system is (0,0). B' will be (6,6). P' will be (2,2) dividing A'B' by 1:2. So P will be (2-1,2+3) = (1,5).

three: By using scalation / scaling method.  Ratio to divide AB: 1:2 or 1/3 : 2/3
 
         P( x , y )  :    x = -1 * 2/3 + 5 * 1/3  = 3/3 = 1
                           y =  3 * 2/3 + 9 * 1/3 =  15/3 = 5
Answer is (1,5)