Question

please answer this question​

Answer 1

Let us assume that the three non-coincident points A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) are collinear. Then, one of these three points will divide the line segment joining the other two internally in a definite ratio. Suppose, the point B divides the line segment AC internally in the ratio λ : 1.

Hence, we have,

(λx₃ + 1 ∙ x₁)/(λ + 1) = x₂ …..(1)

and (λy₃ + 1 ∙ y₁)/(λ+1) = y₂ ..…(2)

From (1) we get,

λx₂ + x₂ = λx₃ + x₁

or, λ (x₂ - x₃) = x₁ - x₂

or, λ = (x₁ - x₂)/(x₂ - x₃)

Similarly, from (2) we get, λ = (y₁ - y₂)/(y₂ - y₃)

Therefore, (x₁ - x₂)/(x₂ - x₃) = (y₁ -y₂)/(y₂ - y₃)

or, (x₁ - x ₂)(y₂ - y₃) = (y₁ - y₂) (x₂ - x₃ )

or, x₁ (y₂ - y₃) + x₂ y₃ - y₁) + x₃ (y₁ - y₂) = 0

which is the required condition of collinearity of-the three given points.

Answer 2

Answer:

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