If $C$ is a point lying on the line segment $A B$ joining $A (1, 1)$ and $B (2, - 3)$ such that $3 A C = C B$ , then find the coordinates of $C$ .

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Solution

Using the section formula, if a point $(x, y)$ divides the line joining the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ in the ratio $m : n$ , then
$(x, y) = (\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n})$
$3 A C = C B$
$\frac{A C}{C B} = \frac{1}{3}$
Since, $A (1, 1)$ and $B (2, 3)$ & $m : n = 1 : 3$
Therefore, we have
$C (\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n})$
$(\frac{(1) (2) + 3 (1)}{1 + 3}, \frac{(1) (3) + 3 (1)}{1 + 3})$
$C \to (\frac{5}{4}, \frac{3}{2})$ .

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