Question:
By using the concept of equation of a line, prove that the three points (3, 0), (–2, –2) and (8, 2) are collinear.
Solution:
In order to show that points (3, 0), (–2, –2), and (8, 2) are collinear, it suffices to show that the line passing through points (3, 0) and (–2, –2) also passes through point (8, 2).
The equation of the line passing through points (3, 0) and (–2, –2) is
$(y-0)=\frac{(-2-0)}{(-2-3)}(x-3)$
$y=\frac{-2}{-5}(x-3)$
$5 y=2 x-6$
i.e., $2 x-5 y=6$
It is observed that at $x=8$ and $y=2$,
L.H.S. $=2 \times 8-5 \times 2=16-10=6=$ R.H.S.
Therefore, the line passing through points (3, 0) and (–2, –2) also passes through point (8, 2). Hence, points (3, 0), (–2, –2), and (8, 2) are collinear.