Question

The locus of the complex number $z$such that \[\arg \left( {\dfrac{{z - 2}}{{z + 2}}} \right) = \dfrac{\pi }{3}\] is:
$
  A)\,A\,\,circle \\
  B)\,A\,\,straight\,\,line \\
  C)\,A\,\,parabola \\
  D)\,An\,\,ellipse \\
$

Answer 1

Hint: Use the properties of arguments and solve it.
We are going to use properties of arguments which is$\arg \left( {\dfrac{{{z_1}}}{{{z_2}}}} \right) = \arg ({z_1}) - \arg ({z_2})$ and then we consider two points $ - 2$ and $2$ and$z$can be anywhere on the arc formed between the two considered point. Then we can consider the argument $z$ at anywhere the arc it forms \[\dfrac{\pi }{3}\] at the point z anywhere on the arc between the given two points $ - 2$ and $2$.So, we can come to a conclusion that the locus of the given complex number is that of a circle.

Complete step by step answer:
 We are given an argument which is
\[\arg \left( {\dfrac{{z - 2}}{{z + 2}}} \right) = \dfrac{\pi }{3}\]
Then, we know the properties of arguments that can be used for the given argument such that it can be understandable.
The property which will be used is$\arg \left( {\dfrac{{{z_1}}}{{{z_2}}}} \right) = \arg ({z_1}) - \arg ({z_2})$
Then on applying on the given argument, we get
$\arg \left( {\dfrac{{z - 2}}{{z + 2}}} \right) = \arg (z - 2) + \arg (z + 2)$
So, let us consider two points $ - 2$ and $2$, then the draw an arc, then we can say that the given argument lies between these two points and anywhere on the drawn arc forming an angle of \[\dfrac{\pi }{3}\] between the given two points forming the angle at $z$,from this we can come to a conclusion that the locus of the given complex number is a circle.

So, the correct answer is Option A.

Note: We have considered the two points according to the constants present in the given arguments and they are not always present around the present considered point as the locus of the complex number can be representing any other figure from the given options.