Find all the complex numbers z such that w = 2 z − 1 z − 2 has argument π 2

Question

Find all the complex numbers   z such that   w  =            2      z      −      1              z      −      2       has argument       π    2

Nelson Santana · Accepted Answer

w  =            (      2      x      −      1      +      2      y      i      )      (      x      −      2      −      y      i      )              (      x      −      2              )        2            +              y        2              ,which gives  (  2  x  −  1  )  (  x  −  2  )  +  2      y    2    =  0and  −  (  2  x  −  1  )  y  +  2  (  x  −  2  )  y  &amp;gt;  0.There are infinitely many these numbers.

Koronicaqn · Accepted Answer

You can solve it geometrically: consider the Argand-Cauchy plane. The condition is equivalent to

Arg \frac{z - \frac{1}{2}}{z - 2} = \frac{π}{2},

and the set of images of

z

which satisfy this relation is the circle with diameter the points of the real axis

\frac{1}{2}

and

2

, these points being removed.

Find all the complex numbers z such that w=(2z-1)/(z-2) has argument pi/2

Answered question

Answer & Explanation

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