227.
A line segment joining the points P(3, 3) and Q(6, - 6) is trisected at the points A and B such that A is nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.
Let the point A(x, y) divide the line segment joining the points P(3, 3) and Q(6, - 6) in the ratio 1 : 2. Then,

Now, the co-ordinates of P are given as
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Here, we have, x1 = 3, y1 = 3;
x2 = 6, y2 = -6
So, co-ordinate of A be,

∴ The point A is (4, 0) which lies on the line
2x + y + k = 0
⇒ 2(4) + 0 + k = 0
⇒ k = - 8. Ans.
Problems Based on Area of Triangle
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