Question

If the points $(2,3),(3,5)$ and $(5,k)$ are collinear, then the value of $k$ is

Answer 1

If the points are collinear, then the area formed by the three points is $0$, so
$\frac{1}{2}\left|\begin{array}{cc}{x}_1& y_1\\ x_2& y_2\\ x_3& y_3\\ x_1& y_1\end{array}\right|=0$

$\Rightarrow \frac{1}{2}\left|\begin{array}{cc}2& 3\\ 3& 5\\ 5& k\\ 2& 3\end{array}\right|=0$

$\Rightarrow |(10-9)+(3k-25)+(15-2k)|=0\Rightarrow |1+3k-25+15-2k|=0\Rightarrow k=9$