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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)`.
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Solution
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
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