Question

the complex number
satisfying arg(z+1)=π/4 and arg(2z+3-2i)=3π/4 simultaneously is?​

Answer 1

Given: complex number z satisfying arg(z+1)=π/4 and arg(2z+3-2i)=3π/4 simultaneously

To Find : z

Solution:

z = x + iy

z + 1  = x + 1 + iy

arg(z+1)=π/4   tanπ/4  = 1

=> x + 1 = y

=> x = y - 1

2z + 3 - 2i

= 2x + 2yi +  3- 2i

= (2x + 3)  + i(2y - 2)

arg(2z+3-2i)=3π/4

tan 3π/4  = - 1

=> 2x + 3  = -(2y - 2)

=> 2x + 2y  = - 1

=> 2(y - 1) + 2y = - 1

=> 4y  =  1

=> y = 1/4

x = 1/4 - 1 = - 3/4

z =  -3/4  + 1/4 i

=>  z =  ( -3 + i) / 4

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