Question

For what values of k are the points (8, 1), (3, –2k) and (k, –5) collinear ?

Answer 1

\[\therefore \frac{1}{2}\left[ 8\left( - 2k + 5 \right) + 3\left( - 5 - 1 \right) + k\left( 1 + 2k \right) \right] = 0\]
\[ \Rightarrow - 16k + 40 - 18 + k + 2 k^2 = 0\]
\[ \Rightarrow 2 k^2 - 15k + 22 = 0\]
\[ \Rightarrow 2 k^2 - 11k - 4k + 22 = 0\]
\[ \Rightarrow k\left( 2k - 11 \right) - 2\left( 2k - 11 \right) = 0\]
\[ \Rightarrow \left( k - 2 \right)\left( 2k - 11 \right) = 0\]
\[ \Rightarrow k - 2 = 0 or 2k - 11 = 0\]
\[ \Rightarrow k = 2 or k = \frac{11}{2}\]

Thus, the values of k are 2 and

\[\frac{11}{2}\]