Find all the non zero complex numbers z satisfy
z bar = iz²
Answers
Solution
Let z = x + iy
_
z. = iz²
_
putting the value of z and z
we know that z bar = x - iy
x - iy = i ( x + iy)²
x - iy = i(x² + i²y² + 2ixy )
(i² = -1 )
x - iy = i( x² - y² +2ixy )
x - iy = ix² - iy² -2xy
x + 2xy - iy - ix² + iy² = 0
Taking i common
x + 2xy - i(x² - y² + y ) = 0
x + 2xy = 0. (i)
x² - y² + y = 0. (ii)
From eq ( i )
x + 2y = 0
x ( 1 + 2y ) = 0
x. = 0/ 1+ 2y
x. = 0
Or ,
x ( 1+ 2y ). = 0
1+ 2y. = 0
2y. = -1
y. = -1/2
Case 1
When x = 0
Putting x = 0 in eq ( ii )
- y² + y. = 0
y ( -y +1). = 0
y ( y - 1 ). = 0
y = 0
or ,
y - 1. = 0
y. = 1
Thus we have value of x and y as
z = 0 + i0
z = 0 + 1i
Case 2
When y = - 1/2
Putting y = - 1/2
x² - y² + y = 0
x² -1/4 - 1/2 = 0
x² -3/4. = 0
x. = +- √3/2
Thus we have values of x and y as
z = √3/2 -1/2 i
z = - √3/2 - 1/2 i
Hope it helps