Point P divides the lne segment joning the points A (2,1) and `B(5,-8)` such that `(AP)/(AB)=1/3` . If P lies on the line ` 2x - y + k = 0 ,` find the value of k

Answer

Values of k is -8

Explanation

`(AP)/(AB) = 1/3`
` :. 3 AP = AB`
` 3AP - AP = PB " " …(A-P-B)`
` :. 2AP = PB`
` (AP)/(PB) = 1/2`
` :. ` P divides the segment AB in the ratio AP : PB
i.e 1:2
A (2,1) and B `(5,-8)`
Let `A(x_(1),y_(1))and B (x_(2),y_(2)) and P (x,y)`
` :. x_(1) = 2 , y_(1) = 1 , x_(2) = 5 and y_(2) = -8`
` m : n = 1 :2`
By section formula ,
` x = (mx_(2)+nx_(1))/ (m+n) " " and y = (my_(2)+ny_(1))/(m+n)`
` :. x = (1(5)+2(2))/(1+2) " " :. y = (1(-8)+2(1))/(1+2)`
` :. x = 9/3 " " :. y = (-6)/3 `
` :. x = 3 " " :. y = -2`
` :. ` the coordinates of point P will be `(3,-2)`
`P (3,-2) " lies on the line " 2x - y + k = 0`
` :. ` its coordinates satisfy the equation of the line substituting x = 3
and ` y = -2 "in " 2x - y +k = 0,` we get
` 2(3) -(-2) +k =0`
` :. 6+ 2+ k =0`
` :. 8+k = 0`
` :. k = -8`