Question

Show that the points (a + 5, a - 4), (a - 2, a + 3) and (a, a) do not lie on a straight line for any value of a.

Answer 1

Given, the points are (a + 5, a - 4), (a-2, a+3) and (a,a)
$\therefore \Delta =\frac{1}{2}\left|\begin{array}{ccc}a+5& a-4& 1\\ a-2& a+3& 1\\ a& a& 1\end{array}\right|$
$=\frac{1}{2}\left|\begin{array}{ccc}5& -4& 0\\ -2& 3& 0\\ a& a& 1\end{array}\right|$ $[\because R_1\to R_1-R_{3}and{R}_2\to R_2-R_3]$
$=\frac{1}{2}\left[1\right(15-8\left)\right]\Rightarrow =\frac{7}{2}\ne 0$
Hence, given points form a triangle i.e., points do not lie in a straight line.