SOLUTION: The points representing the complex number z for which the arg{{{ (z - 2)/(z + 2) = pi/3 }}} lie on which of the following: A- Circle B- Straight Line C- Ellipse D- Parabola [

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: The points representing the complex number z for which the arg{{{ (z - 2)/(z + 2) = pi/3 }}} lie on which of the following: A- Circle B- Straight Line C- Ellipse D- Parabola [      Log On


   

Question 1184304: The points representing the complex number z for which the arg+%28z+-+2%29%2F%28z+%2B+2%29+=+pi%2F3+ lie on which of the following:
A- Circle
B- Straight Line
C- Ellipse
D- Parabola
[Note: Here arg means argument and pi means the value 3.142...]
Found 3 solutions by MathLover1, ikleyn, robertb:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

correct option is: A. circle

Given that:
arg%28z-2%29%2Farg%28z%2B2%29=pi%2F3
arg%28z-2%29-arg%28z%2B2%29=+pi%2F3
Hence, two fixed points (-2,0) and (2,0) subtend an angle of pi%2F3 at z.
So, z lies on major+arc of a circle whose chord+is the line joining (-2,0) and (2,0) as shown in the figure belowe.
198767-14600-ans-51e9b15dada14e13a7299672a793ec89-1

Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.

In this problem we have two selected points in the complex plane: A = (-2,0) 
and B = (2,0).


The complex number z-2 represents a vector connecting point z with the point (2,0).


The complex number z+2 represents a vector connecting point z with the point (-2,0).


Since  arg%28%28z-2%29%2F%28x%2B2%29%29 = pi%2F3  (given), it means that the angle between the vectors  (z-2)  and  (z+2)  is  pi%2F3.


It means that the point z lies on the arc of a circle and the vectors (z-2) and (z+2)  form an inscribed angle  

of the measure of  pi%2F3 = 60 degrees, which leans on the segment AB as on the chord.



The locus is shown in the Figure below in red line.


    The parts of the circles shown in black are not the parts of the locus.



Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
===> Let z be a complex number satisfying the given condition.
,

where the subscript "c" denotes conjugation.
===> = alpha+%2B+i%2A%28beta%29.

The given condition then says that


===> sqrt%283%29=+%284%2AIm%28z%29%29%2F%28abs%28z%29%5E2+-4%29, assuming that abs%28z%29 doesn't lie on the circle abs%28z%29+=+2.

===> abs%28z%29%5E2+=+%284%2Fsqrt%283%29%29%2AIm%28z%29%2B4, after solving for abs%28z%29%5E2.

If we let z = x + i*y, then the preceding equation says that

red%28x%5E2+%2B+y%5E2+=+%284%2Fsqrt%283%29%29%2Ay+%2B+4%29 <----this circle!

which is clearly an equation of a circle. Therefore the answer is option highlight%28A%29.
Whether anyone denies this or not is immaterial, because the z values satisfying +arg%28%28z+-+2%29%2F%28z+%2B+2%29%29+=+pi%2F3+ still red%28LIES%29 red%28ON%29 this circle.
It doesn't lie on a straight line, not on an ellipse, not ona parabola, but on a CIRCLE.

And even if z is only on a small part of the circle doesn't matter, because IT IS STILL ON THE CIRCLE.