the line segment joining the points A ( 2 , 1 ) B ( 5 , -8 ) is trisected at the points p and q such that P is nearer to A . if P also lies on the line given by 2 x - y + k =0 , find the value of k.

The given point are A(2, 1) and B(5, – 8).

Given, P and Q trisects the line segment AB.

∴ AP = PQ = QB

PB = PQ + QB = AP + AP = 2AP

AP : PB = AP : 2AP = 1 : 2

∴ P divides the line segment AB in the ration 1 : 2.

∴ Coordinates of P

P(3, – 2) lies on 2*x* – *y* + *k* = 0.

∴ 2 × 3 – (– 2) + *k* = 0

⇒ 6 + 2 + *k* = 0

⇒ 8 + *k* = 0

⇒ *k* = – 8

Thus, the value of *k* is – 8.

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