Find the ratio in which the line `2x+3y-5=0` divides the line segment joining the points (8,-9) and (2,1). Also find the coordinates of the points of division.

Explanation

Let the line `2x+3y-5=0` divides the line segment joining the points A (8,-9) and B (2,1) in the ratio `lambda:1` at point P.
`:.` Coordinates of P `-={(2lambda+8)/(lambda+1),(lambda-9)/(lambda+1)}`
[`:'` internal division `={(m_(1)x_(2)+m_(2)x_(1))/(m_(1)+m_(2)),(m_(1)y_(2)+m_(2)y_(1))/(m_(1)+m_(2))}]`
But P lies on `2x+3y-5=0`.
`:. 2((2lambda+8)/(lambda+1))+3((lambda-9)/(lambda+1))-5=0`
`rArr2(2lambda+8)+3(lambda-9)-5(lambda+1)=0`
`rArr4lambda+16+3lambda-27-5lambda-5=0`
`rArr2lambda-16=0`
`rArrlambda=8lambdarArrlambda :1=8 : 1`
So , the point P divised the line in the ratio `8 : 1` .
`:.` Point of division P `-={(2(8)+8)/(8+1),(8-9)/(8+1)}`
`-=((16+8)/(9),-(1)/(9))`
`-=((24)/(9),(-1)/(9))-=((8)/(3),(-1)/(9))`
Hence , the required point of division is `((8)/(3),(-1)/(9))`.