Question

If z₁ and z₂ both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.

Answer 1

Let z = x + iy, z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ .

Then `z + barz = 2|z - 1|`

⇒ (x + iy) + (x – iy) = `2|x - 1 + "i"y|`

⇒ 2x = 1 + y²    .......(1)

Since z₁ and z₂ both satisfy (1), we have

`2x_1 = 1 + y_1^2 .....` and `2x_2 = 1 + y_2^2`

⇒ `2(x_1 - x_2) = (y_1 + y_2)(y_1 - y_2)`

⇒ 2 = `(y_1 + y_2) ((y_1 - y_2)/(x_1 - x_2))`  ......(2)

Again `z_1 - "z"_2 = (x_1 - x_2) + "i"(y_"i" - y_2)`

Therefore, tanθ = `(y_1 - y_2)/(x_1 - x_2)`, where θ = arg`("z"_1 - "z"_2)`

⇒ `tan  pi/4 = (y_1 - y_2)/(x_1 - x_2)`  ......`("Since"  theta = pi/4)`

i.e., 1 = `(y_1 - y_2)/(x_1 - x_2)`

From (2), We get 2 = y₁ + y₂ i.e., `"Im" ("z"_1 + "z"_2)` = 2