Question

What is the solution of the equation $\left| z \right|-z=1+2i$?
A. $\left( \dfrac{3}{2} \right)+2i$
B. $\left( \dfrac{3}{2} \right)-2i$
C. $3-2i$
D. None of these

Answer 1

Hint: Complex numbers always have real and imaginary parts. Therefore, substituting the values and equating the real and imaginary part will give you the value of z. You should consider z as the sum of real and imaginary numbers. Let $z=x+iy$ and equate the real and imaginary part.

Complete step by step solution: 
Let $z=x+iy$ and substituting the values in question one by one:
First we have to find the magnitude of complex number z.
The magnitude of the complex number z is
\[\left| z \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}\]
Now from the question we can write,
\[\sqrt{{{x}^{2}}+{{y}^{2}}}-x-iy=1+2i\]
Equating real and imaginary parts of the equation we get
First equating the imaginary part we get the value of y.
$\therefore y=-2$
Now equating real part of the equation we get:
$\sqrt{{{x}^{2}}+{{y}^{2}}}=1+x$
Squaring both sides,
${{x}^{2}}+{{y}^{2}}=1+{{x}^{2}}+2x$
We have the value of y=-2, on substituting it to the equation
Then the value of $x=\dfrac{3}{2}$
Substituting the obtained values of y and x in z:
$\therefore z=\dfrac{3}{2}-2i$
Therefore, the Answer is option (b)

Notes: When we do the problems related to complex numbers you should take care of signs while equating equations otherwise there is a high probability of getting the wrong answer. Also you should know the minimum facts that a complex number z occurs in the form $z=x+\text{i}y$. x and y are real numbers and ‘i’ is a symbol showing imaginary part and the value of ‘i’ is ${{\text{i}}^{2}}=-1$. The symbol ‘i’ is generally called an imaginary unit. In this form of complex number, x is known as the real part of the equation and y is the imaginary part of the solution. We have to note that the real and imaginary parts are always real numbers.