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Solve z+|z|=2+8i to Find |z| | Complex Numbers JEE

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

Question

If the complex number

z

satisfies

z + |z| = 2 + 8i

, then the value of

|z|

will be

8

17

correct

15

24

Solution

Step 1: Write

z = a + ib

, so

|z| = \sqrt{a^2 + b^2}

(a real number). The equation becomes:

a + ib + \sqrt{a^2 + b^2} = 2 + 8i

Step 2: Separate into real and imaginary parts. The imaginary part of the left side is just

b

(the modulus is real), so:

\text{Imaginary: } b = 8

\text{Real: } a + \sqrt{a^2 + b^2} = 2

Step 3: Substitute

b = 8

into the real equation:

a + \sqrt{a^2 + 64} = 2 \implies \sqrt{a^2 + 64} = 2 - a

Step 4: Square both sides (valid since the left side is non-negative, which also requires

2 - a \geq 0

a^2 + 64 = (2 - a)^2 = 4 - 4a + a^2

64 = 4 - 4a \implies 4a = -60 \implies a = -15

(The requirement

2 - a \geq 0

holds since

2 - (-15) = 17 > 0

.) Step 5: Compute the modulus:

|z| = \sqrt{a^2 + b^2} = \sqrt{(-15)^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17

Correct answer: (2)

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