Question

Locate the complex numbers $z = x + iy$ for which ${\log _{\sqrt 3 }}\dfrac{{|z{|^2} - |z| + 1}}{{2 + |z|}} < 2$.

Answer 1

According to the question, complex numbers are any number that can be written in the form of $a + bi$, where ‘i’ is the imaginary unit and ‘a’ and ‘b’ are real numbers. Here, ‘i’ is the symbol of the imaginary unit and it satisfies the equation ${i^2} = - 1$.

Complete step-by-step solution:
A simple property of logarithm is being used here that is:
${\log _{{a^{}}}}b = c$ then ${a^c} = b$
So, here in this equation, according to the complex form, $a$ is represented by $\sqrt 3$, $b$ is represented by $\dfrac{{\left| {{z^2}} \right| - \left| z \right| + 1}}{{2 + \left| z \right|}}$, and $c$ is represented by $2$.
So therefore, $\dfrac{{\left| {{z^2}} \right| - \left| z \right| + 1}}{{2 + \left| z \right|}} < {\sqrt {{3^{}}} ^2}$
Now, when we solve this, we get:

\Rightarrow \dfrac{{\left| {{z^2}} \right| - \left| z \right| + 1}}{{2 + \left| z \right|}} < {3^{2 \times \dfrac{1}{2}}} \\\ \Rightarrow \dfrac{{\left| {{z^2}} \right| - \left| z \right| + 1}}{{2 + \left| z \right|}} < 3 \\\ \Rightarrow \left| {{z^2}} \right| - \left| z \right| + 1 < 3(2 + \left| z \right|) \\\ \Rightarrow \left| {{z^2}} \right| - \left| z \right| + 1 < 6 + 3\left| z \right| \\\ \Rightarrow \left| {{z^2}} \right| - 4\left| z \right| + ( - 5) < 0 $$ Now we will try to split the equation, and we get: $$ \Rightarrow \left| {{z^2}} \right| + \left| z \right| - 5\left| z \right| - 5 < 0$$ Now, we will take out the common terms, and we get: $$ \Rightarrow \left| z \right|(\left| z \right| + 1) - 5(\left| z \right| + 1) < 0 \\\ \therefore (\left| z \right| - 5)(\left| z \right| + 1) < 0 \\\ \therefore - 1 < \left| z \right| < 5 $$ But as we know, $$\left| z \right| > 0$$ $$ \Rightarrow 0 < \left| z \right| < 5$$ Now this represents that all points of $$z$$lie in the circle of radius $$5$$ with centre $$(0,0)$$. We can see in the below diagram,  **Note:** We wrote $$z = x + iy$$, hence it was a cartesian equation of writing a complex number. The value of $$\left| z \right|$$ comes out to be between $$ - 1$$ and $$5$$, but as we take only positive numbers, this value of $$\left| z \right|$$is between $$0$$ and $$5$$. Thus, we make a circle which comprises all the numbers between $$0$$ and $$5$$ as to represent them in the cartesian form.