Question 9
The point P(5,-3) is one of the two points of trisection of line segment joining the points A(7,-2) and B(1,-5).

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Solution

True
Let P(5,-3) divides the line segment joining the points A(7,-2) and B(1,-5) in the ratio K:1 internally.
By section formula, the coordinate of point P will be
$(\frac{k (1) + (1) (7)}{k + 1}, \frac{k (- 5) + 1 (- 2)}{k + 1}) i . e ., (\frac{k + 7}{k + 1}, \frac{- 5 k - 2}{k + 1}) N o w, (5, - 3) = (\frac{k + 7}{k + 1}, \frac{- 5 k - 2}{k + 1}) \Rightarrow \frac{k + 7}{k + 1} = 5 5 k + 5 = k + 7 \Rightarrow - 4 k = - 2 ∴ k = \frac{1}{2}$
So the point P divides the line segment AB in ratio 1:2.
Hence, point P in the point of trisection of AB.

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