Show that the points (a + 5, a – 4), (a – 2, a + 3)

Given points are (a + 5, a – 4), (a – 2, a + 3) and (a, a).

Now, we have to prove that these points do not lie on a straight line.

So, if we prove that these points form a triangle then it can’t line on a straight line.

Area, $\Delta=\frac{1}{2}\left|\begin{array}{ccc}a+5 & a-4 & 1 \\ a-2 & a+3 & 1 \\ a & a & 1\end{array}\right|$

[Applying $R_{1} \rightarrow R_{1}-R_{3}$ and $R_{2} \rightarrow R_{2}-R_{3}$ ]

$=\frac{1}{2}\left|\begin{array}{ccc}5 & -4 & 0 \\ -2 & 3 & 0 \\ a & a & 1\end{array}\right|=\frac{1}{2}\left[(1 \cdot(15-8)]=\frac{7}{2} \neq 0\right.$

Hence, the given points form a triangle and can’t lie on a straight line.