In what ratio does the line joining the points (2, 3) and (4, 1) divide the segment joining the points (1, 2) and (4, 3)?

Explanation

The equation of the line joining the points (2,3) and (4,1) is
`y-3 = (1-3)/(4-2) (x-2)`
or y-3 = -x+2
or x+y-5 = 0
Suppose the line joining (2,3) and (4,1) divides the segment joining (1,2) and (4,3) at point P in the ratio `lambda` : 1.
Then the coordinates of P are
`((4lambda + 1)/(lambda + 1), (3lambda +2)/(lambda + 1))`
Clearly, P lies on (1), Therefore.
`(4lambda + 1)/(lambda + 1) + (3lambda +2)/(lambda + 1) = 5`
` "or " lambda = 1`
Hence, the required ratio is `lambda:1` i.e., 1:1.