Find the value of $K$ , for which the given points are collinear:
$\begin{matrix} A (7, - 2) \end{matrix} \begin{matrix} B (5, 1) \end{matrix} \begin{matrix} C (3, K) \end{matrix}$

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Solution

As the three points are collinear: $A B + B C = C A$ and $A B$ and $A C$ will have the same slope.
Slope of $A B = \frac{- 2 - 1}{7 - 5}$ (1)
Slope of $A B = \frac{- 2 - K}{7 - 3}$ (2)
Equating (1) and (2) gives
$\frac{- 3}{2} = \frac{- 2 - K}{4}$
$3 = \frac{2 + K}{2}$
$6 = 2 + K$
Hence $K = 4$

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