Question

If $z(\overline z + 3) = 2$ then the locus of $z = x + iy$ is
1)${x^2} + {y^2} + 3x - 2 = 0,y = 0$
2)${\text{x = 0}}$ such that ${\text{y > }}\dfrac{2}{3}$
3)\[{{\text{x}}^{\text{2}}}{\text{ + }}{{\text{y}}^{\text{2}}}{\text{ + 2x - 4x = 0 }}\] such that \[{\text{y < 0, }}{x^2} + {y^2} > 1\]
4)${{\text{x}}^{\text{2}}}{\text{ + }}{{\text{y}}^{\text{2}}}{\text{ + 2x - 4y = 0 }}$ such that ${\text{2x - y + 4 > 0}}$

Answer 1

Hint: Here using the complex number concept in this question. Substitute the values in the complex number $z$ . Simplify the equations, we will find the answer for the question in the given option.

Formula using:
$(a + b)(a - b) = {a^2} - {b^2}$
Using this formula for this question.

Complete step-by-step answer:
Here $z = x + iy$ is a Complex Number, so this equation is substituting the question$ - - - - I$
In Above Equation ${\text{x,i}}$ are the real part, $y$ is the imaginary part. These two parts substitute the given question we have found out the answer.
So here $\overline z = x - iy - - - - - II$ this is an equation two these also substitute the question we find the answer
We substitute $I,II$equation value in following equation,
$\Rightarrow$ ${\text{z(}}\overline {\text{z}} + 3) = 2$
Simply the values in double side,
$\Rightarrow$ $z\overline z + 3z = 2$
Substitute the above equation,
$\Rightarrow$ $(x + iy)(x - iy) + 3(x + iy) = 2$
The above equation as same like as a $(a + b)(a - b) = {a^2} - {b^2}$
$
\Rightarrow {x^2} - {i^2}{y^2} + 3x + 3iy - 2 = 0 \\
\Rightarrow {x^2} + {y^2} + 3x - 2 + i(3y) = 0 \\
 $ here I =0
Equating real and imaginary terms,
$\Rightarrow$ ${x^2} + {y^2} + 3x - 2 = 0,3y = 0,y = 0$
Here ${x^2} + {y^2}{\text{ + 3x - 2 = 0 , y = 0 }}$
The above equation is the same as option 1. So, the correct answer is option 1 ${x^2} + {y^2} + 3x - 2 = 0,y = 0$

Additional Information:
Complex number has two parts that are real parts and imaginary parts. So, we find the answer for substituting the two parts in the equation. And the locus values. Here also using another formula for simplifying the answer.

Note: Here, we have a complex number and we will find the locus of the given equation. First, we have a clear idea about the real part and imaginary part and substitute the values in these parts. Another important one is arithmetic operators. It is a very important one. We Know the concept of which operator where we used.