If P (9a-2,-b) divides line segment joining A (3a+1,-3) and B(8a,5) in the ratio `3 : 1`, then find the values of a and b.

Explanation

Let P(9a-2,-b) divides AB internally in the ration `3 : 1`.
By section formula ,
`9a-2=(3(8a)+1(3a+1))/(3+1)`
[`:'` internal section formula , the coordinates of point P divides the line seggment joining the Point `(x_(2),y_(1))and(x_(2),y_(2))` in the ratio `m_(1) :m_(2)` internally is `((m_(2)x_(1)+m_(1)x_(2))/(m_(1)+m_(2)),(m_(2)y_(1)+m_(1)y_(2))/(m_(1)+m_(2)))]`
and `-b=(3(5)+1(-3))/(3+1)`
`rArr9a-2=(24a+3a+1)/(4)`
and `-b=(15-3)/(4)`
`rArr9a-2=(27a+1)/(4)`
and `-b=(12)/(4)`
`rArr36a-27a-8-1=0`
`rArr9a-9=0`
`:.a=1` Hence , the required values of a and b are 1 and -3.