Question

#Quality Question

@Complex Numbers

Solve the equation

$2z = |z| + 2 i$



​

Answer 1

$\large\underline{\sf{Solution-}}$

Since, z is a complex number

Let us assume that z = x + iy

According to statement,

$\rm :\longmapsto\:2z = |z| + 2i$

On substituting the value of z, we get

$\rm :\longmapsto\:2(x + iy) = |x + iy| + 2i$

We know, that if z = x + iy, then

$\rm :\longmapsto\: |z| = \sqrt{ {x}^{2} + {y}^{2} }$

On substituting the value, we get

$\rm :\longmapsto\:2x + 2iy = \sqrt{ {x}^{2} + {y}^{2} } + 2i$

On comparing, imaginary parts, we get

$\rm :\longmapsto\:2y = 2$

$\bf\implies \:y = 1$

Now, On comparing Real part, we get

$\rm :\longmapsto\:2x = \sqrt{ {x}^{2} + {y}^{2} }$

On substituting the value of y = 1, we get

$\rm :\longmapsto\:2x = \sqrt{ {x}^{2} + {1}^{2} }$

$\rm :\longmapsto\:2x = \sqrt{ {x}^{2} + 1 }$

On squaring both sides, we get .

$\rm :\longmapsto\: {4x}^{2} = {x}^{2} + 1$

$\rm :\longmapsto\: {3x}^{2} = 1$

$\rm :\longmapsto\: {x}^{2} = \dfrac{1}{3}$

$\bf\implies \:x = \: \pm \: \dfrac{1}{ \sqrt{3} }$

Justification of solution,

When

$\bf :\longmapsto\: z = - \: \dfrac{1}{ \sqrt{3} } + i$

Then,

$\bf :\longmapsto\: = - \: \dfrac{2}{ \sqrt{3} } + 2i = \sqrt{\dfrac{1}{3} + 1 } + 2i$

which is not satisfied as Real part on LHS is negative and Real part is positive.

When

$\bf :\longmapsto\: z = \: \dfrac{1}{ \sqrt{3} } + i$

then

$\bf :\longmapsto\: \: \dfrac{2}{ \sqrt{3} } + 2i = \sqrt{\dfrac{1}{3} + 1 } + 2i$

Satisfied

Hence,

Solution of

$\rm :\longmapsto\:2z = |z| + 2i$

is

$\bf :\longmapsto\: z = \: \dfrac{1}{ \sqrt{3} } + i$

Answer 2

*Question:-

Solve the equation :→ 2z = |z| + 2i

*Answer:-

$2(x +iy) = \sqrt{ {x}^{2} + {y}^{2} } + 2i$

$2x = \sqrt{ {x}^{2} + {y}^{2} }$

And,

$2y = 2$

i.e. ,

$y = 1$

$4 {x}^{2} = {x}^{2} + 1$

i.e. ,

${3x}^{2} = 1$

i.e. ,

$x = ± \frac{1}{ \sqrt{3} }$

$x = \frac{1}{ \sqrt{3} } \:$

Since,

$x≥0$

Hence,

$z = \frac{1}{ \sqrt{3} } + i = \frac{ \sqrt{3} }{3} + i$

Hope it helps you@BrainlyTurtle.

I like your way of writing answers.