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Let z be a complex number such that $\left | z \right |+z=3+i$ (where $i=\sqrt{-1}$ ). The $\left | z \right |$ is equal to :
Option 1)
$\frac{5}{3}$
Option 2)
$\frac{\sqrt{41}}{4}$
Option 3)
$\frac{5}{4}$
Option 4)
$\frac{\sqrt{34}}{3}$

Answers (1)

best_answer

Definition of Modulus of z(Complex Number) -

$\left | z \right |=\sqrt{a^{2}+b^{2}}$ is the distance of z from origin in Argand plane

- wherein

Real part of z = Re (z) = a & Imaginary part of z = Im (z) = b

$\left | z \right |+z=3+i$

$z=3-\left | z \right |+i.............(1)$

$Let \: a\: =3-\left | z \right |$

$=>\left | z \right |=3-a$ .................(2)

Now eqn ( 1 )

$z=a+i$

$=>\left | z \right |=\sqrt{a^{2}+1}$ ...............(3)

Using eqn ( 2 ) and ( 3 )

$3-a=\sqrt{a^{2}+1}$

$(3-a)^{2}={a^{2}+1}$

$a=\frac{4}{3}$

$\left | z \right |=3-a=\frac{5}{3}$

Option 1)
$\frac{5}{3}$
Option 2)

$\frac{\sqrt{41}}{4}$

Option 3)

$\frac{5}{4}$

Option 4)

$\frac{\sqrt{34}}{3}$

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