Show by section formula that the points (3, -2), (5, 2) and (8, 8) are collinear.

Let the point (5, 2) divides the line joining the points (3, -2) and (8, 8) in the ratio m₁ : m₂.

By section formula the x-coordinate of dividing points is given by

$\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}\Big)

Putting values we get,

$5 = \dfrac{m$1 \times 8 + m2 \times 3}{m1 + m2} \\[1em] \Rightarrow 8m1 + 3m2 = 5m1 + 5m2 \\[1em] \Rightarrow 8m1 - 5m1 = 5m2 - 3m2 \\[1em] \Rightarrow 3m1 = 2m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{2}{3}. \qquad \text{….[Eq 1]}

By section formula the y-coordinate of dividing points is given by

$\Big(\dfrac{m$1y2 + m2y1}{m1 + m2}\Big)

Putting values we get,

$2 = \dfrac{m$1 \times 8 + m2 \times (-2)}{m1 + m2} \\[1em] \Rightarrow 8m1 - 2m2 = 2m1 + 2m2 \\[1em] \Rightarrow 8m1 - 2m1 = 2m2 + 2m2 \\[1em] \Rightarrow 6m1 = 4m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{4}{6} \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{2}{3} \qquad \text{….[Eq 2]}

Since, we get ratio m₁ : m₂ from both the equations it means that the point (5, 2) lies on the line joining the points (3, -2) and (8, 8).

Hence, proved that the points (3, -2), (5, 2) and (8, 8) are collinear.