If the points (1, 4), (3, -2) and (p, -5) lie on a line, find the value of p.

Equation of line passing through (1, 4) and (3, -2) i.e two points can be given by two-point formula i.e.

$\Rightarrow y - y$1 = \dfrac{y2 - y1}{x2 - x1}(x - x1) \\[1em] \therefore y - 4 = \dfrac{-2 - 4}{3 - 1}(x - 1) \\[1em] \Rightarrow y - 4 = \dfrac{-6}{2}(x - 1) \\[1em] \Rightarrow y - 4 = -3(x - 1) \\[1em] \Rightarrow y - 4 = -3x + 3 \\[1em] \Rightarrow y = -3x + 7.

Since (p, -5) lies on the line it will satisfy it. Putting the values,

⇒ -5 = -3p + 7
⇒ 3p = 7 + 5
⇒ 3p = 12
⇒ p = 4.

Hence, the value of p is 4.