By using the concept of the equation of the straight line, prove that the given three points are collinear.

(i) (4, 2), (7, 5) and (9, 7)

(ii) (1, 4), (3, –2) and (–3, 16)

The equation of the line passing through the two given (x₁, y₁) and (x₂,y₂)points is given by form

(i) The equation of the line passing through (4, 2), (7, 5)is

⇒ y–2 = x–4

⇒ x – y –2 = 0

Substituting the 3^rd point in the equation of the line obtained

x–2 = y

⇒ 7 = 9–2

⇒ 7 = 7

Hence LHS = RHS

Since the third point satisfies the equation of the line obtained

⇒ All the three points lies in the same straight line or are collinear.

(ii) (1, 4), (3, –2) and (–3, 16)

The equation of the line passing through the points (1, 4), (3, –2) is

⇒

⇒ y–4 = –3x + 3

⇒ 3x + y –7 = 0

Substituting the 3^rd point in the equation of line obtained

y = –3x + 7

⇒ 16 = –3 (–3) + 7

⇒ 16 = 16

LHS = RHS

Thus the three points are in the same straight line or are collinear.