# If z1 and z2 both satisfy z+z¯=2|z-1| arg(z1-z2)=π4, then find ImIm(z1+z2). - Mathematics

Sum

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).

#### 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

Concept: Concept of Complex Numbers
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 5 Complex Numbers and Quadratic Equations
Solved Examples | Q 15 | Page 83

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