Question

Point P divides the line segment joining 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 1

We have section formula to answer this.
The section formula is attached to this answer.
Variables in the formula: 
This is applicable if A=(x1,y1) and B=(x2,y2) and P is such that AP/PB=m/n.
Solution to your question:
Given that AP/AB=1/3
AB=3AP
AP+PB=3AP
PB=2AP
AP/PB=1/2
This means here m=1 and n=2
Now we can find P as we know all the variables.
x1=2
y1=1
x2=5
y2=-8
m=1
n=2
Now you may find the point P(x,y) by using that formula.
After substituting values, we get 
x=3 and y=-2
Hence P=(3,-2)
Given that P lies on 2x-y+k=0
This means the coordinates of P satisfy the equation 2x-y+k=0
Substitute x=3 and y=-2 in this equation.
2(3)-(-2)+k=0
6+2+k=0
8+k=0
k=-8
Hence the value of k is -8.