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Value of |z+z̄| When |z|=min{|z−1|,|z+1|} | JEE

JEE Maths question with a full step-by-step solution.

Question

|z| = \min\{|z - 1|,\, |z + 1|\}

, then the value of

|z + \bar{z}|

1

correct

2

3

4

Solution

Step 1: Simplify the target quantity. With

z = x + iy

, we have

z + \bar{z} = 2x = 2\,\mathrm{Re}(z)

, so:

|z + \bar{z}| = |2x| = 2|x|

Step 2: Interpret the condition

|z| = \min\{|z - 1|, |z + 1|\}

. The minimum equals one of the two distances, so either

|z| = |z - 1|

|z| = |z + 1|

(whichever is smaller); both extreme cases occur on the boundary where the chosen distance is achieved. Step 3: Solve

|z| = |z - 1|

(distance from origin equals distance from

1

). This is the perpendicular bisector of

0

and

1

x^2 + y^2 = (x - 1)^2 + y^2 \implies 0 = -2x + 1 \implies x = \frac{1}{2}

Step 4: Solve

|z| = |z + 1|

(distance from origin equals distance from

-1

x^2 + y^2 = (x + 1)^2 + y^2 \implies 0 = 2x + 1 \implies x = -\frac{1}{2}

Step 5: In either case

|x| = \dfrac{1}{2}

, so:

|z + \bar{z}| = 2|x| = 2\cdot\frac{1}{2} = 1

Correct answer: (1) Note: This question can solve be by triangle inequality.

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