Question

If the points (x₁, y₁), (x₂, y₂) and (x₁ + x₂, y₁ + y₂) are collinear, then x₁y₂ is equal to ______.

विकल्प

• x₂y₁

• x₁y₁

• x₂y₂

• x₁x₂

Answer 1

If the points (x₁, y₁), (x₂, y₂) and (x₁ + x₂, y₁ + y₂) are collinear, then x₁y₂ is equal to x₂y₁.

Explanation:

Method 1:

When the points (x₁, y₁), (x₂, y₂) and (x₁ + x₂, y₁ + y₂) are collinear in the Cartesian plane, then

`|(x_1 - x_2,y_1 - y_2),(x_1 - (x_1 + x_2),y_1 - (y_1 + y_2))|` = 0

(x₁ − x₂) [y₁ − y₁ − y₂] − (y₁ − y₂) [x₁ − x₁ − x₂] = 0

(x₁ − x₂) (−y₂) − (y₁ − y₂) (−x₂) = 0

−y₂x₁ + x₂y₂ + y₁x₂ − y₂x₂ = 0

−y₂x₁ + y₁x₂ = 0

y₁x₂ = y₂x₁

Method 2:

When the points (x₁, y₁), (x₂, y₂) and (x₁ + x₂, y₁ + y₂) are collinear in the Cartesian plane, then

`|(x_1,y_1,1),(x_2,y_2,1),(x_1 + x_2,y_1 + y_2,1)|` = 0

⇒ 1.(x₂y₁ + x₂y₂ − x₁y₂ − x₂y₂) −1(x₁y₁ + x₁y₂ − x₁y1 − x₂y₁) +(x₁y₂ − x₂y₁) = 0

⇒ x₂y₁ = x₁y₂