Question

If A, B, C are three points in argand plane representing the complex numbers z₁, z₂ and z₃ such that, z₁ = ${\displaystyle \frac{\lambda z_2+z_3}{\lambda +1}}$, where λ ∈ R, then find the distance of point A from the line joining points B and C.

Options

• 0

• λ

• λ + 1

• ${\displaystyle \frac{\lambda }{(\lambda +1)}}$

Answer 1

0

Explanation:

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

Given, z₁ = ${\displaystyle \frac{\lambda z_2+z_3}{\lambda +1}}$

${\displaystyle \Rightarrow }$ x₁ + iy₁ = ${\displaystyle \frac{\lambda (x_2+i{y}_2)+(x_3+i{y}_3)}{\lambda +1}}$

${\displaystyle \Rightarrow }$ x₁ + iy₁ = ${\displaystyle \frac{\lambda x_2+\lambda i{y}_2+x_3+i{y}_3}{\lambda +1}}$

${\displaystyle \Rightarrow }$ x₁ + iy₁ = ${\displaystyle \frac{\lambda x_2+x_3}{\lambda +1}+\frac{i(\lambda y_2+y_3)}{\lambda +1}}$

∴ x₁ = ${\displaystyle \frac{\lambda x_2+x_3}{\lambda +1}}$ and y₁ = ${\displaystyle \frac{\lambda (y_2+y_3)}{\lambda +1}}$

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.