Question

If A, B, C are three points in argand plane representing the complex numbers z₁, z₂ and z₃ such that, z₁ = `(λz_2 + z_3)/(λ + 1)`, where λ ∈ R, then find the distance of point A from the line joining points B and C.

विकल्प

• 0

• λ

• λ + 1

• `λ/((λ + 1))`

Answer 1

0

Explanation:

Let z₁ = x₁ + iy₁, z₂ = x₂ + iy₂ and z₃ = x₃ + iy₃

Given, z₁ = `(λz_2 + z_3)/(λ + 1)`

`\implies` x₁ + iy₁ = `(λ(x_2 + iy_2) + (x_3 + iy_3))/(λ + 1)`

`\implies` x₁ + iy₁ = `(λx_2 + λiy_2 + x_3 + iy_3)/(λ + 1)`

`\implies` x₁ + iy₁ = `(λx_2 + x_3)/(λ + 1) + (i(λy_2 + y_3))/(λ + 1)`

∴ x₁ = `(λx_2 + x_3)/(λ + 1)` and y₁ = `(λ(y_2 + y_3))/(λ + 1)`

Hence, z₁ (point A) divides z₂ (point B) and z₃ (point C) in the ratio λ : 1.

∴ B, A, and C are collinear.

So, the distance of point A from the line joining points B and C is zero.