SOLUTION: If {{{ (z-1)/(z+1) }}} is purely imaginary then what is value of |z|?

Algebra.Com
Question 1184217: If is purely imaginary then what is value of |z|?
Found 2 solutions by robertb, ikleyn:
Answer by robertb(5830)   (Show Source): You can put this solution on YOUR website!
If is purely imaginary, then for some .
Let z = a + bi.
===> z - 1 = A(z+1)i ===> a + bi - 1 = A(a + bi + 1)i
===> a + bi - 1 = A((a + 1)i - b) = A(a+1)i - Ab

<===> (a - 1 + Ab) + (b - A(a+1))i = 0
===> a - 1 + Ab = 0 and b - A(a+1) = 0, or
===> a + Ab = 1 and -Aa + b = A.
===> and after solving the system for a and b.
===>
===>

=

Therefore,


Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.


            This problem has a nice geometric interpretation. 


1 and -1 are selected points in the complex number plane.


For an arbitrary complex number z, the numbers z-1 and z+1 are vectors, connecting z with the points 1 and -1.



The condition that    is purely imaginary means that the vectors z-1 and z+1 are perpendicular.



So, the problem asks to find points z in complex plane, such that the "visibility angle" from z to 

these points "1" and "-1" is the right angle.


Or, in other terms, find points z in complex plane, such that the angle between the vectors z-1 and z+1 is the right angle.



Clearly, these points are on the unit circle, and they provide the right angle leaning on the segment [-1,1]

as on the diameter of the unit circle.   So, |z| = 1.    ANSWER



RELATED QUESTIONS

Prove the proposition |z|=1 if {(1+z)/(1-z)} is purely imaginary. Please... (answered by robertb,richard1234)
If z-2i/2z-1 is purely imaginary prove that the locus of z in the Argands diagram is a... (answered by greenestamps)
if |(z)/(|conj(z)|) -conj(z) | = 1 + | z | , z \[Element] C , then (z imaginary) prove... (answered by Edwin McCravy,ikleyn)
prove that ,if |(z)/(|conj(z)|) -conj(z) | = 1 + | z | , z \[Element] C , then (z... (answered by ikleyn)
If z is a complex number then minimum value of |z| +|z+1|... (answered by stanbon)
x^2=y+z, y^2=z+x, z^2=x+y, then what is the value of (1/(x+1)) + (1/(y+1)) +... (answered by Edwin McCravy)
Find & Sketch the locus of {{{(z-2)/z}}} if it is purely imaginary. Similarily, Find and... (answered by ikleyn)
if w=f(z)=z+1/z.......for what finite values of z is f(z)... (answered by ikleyn)
if x = y+z, y = z+x, z = x+y, then what is the value of (1/(x+1)) + (1/(y+1)) +... (answered by ikleyn)