Question

Find all complex numbers which make the following equations true; |Z+1|=1 |Z^2+1|=1

Answer 1

Step-by-step explanation:

Ur ans :-

Solution:-

Given : Let z=x+iy. Then, z+1=(x+1)+iy

Therefore, ∣z+1∣=

(x+1)

2

+y

2

Now, ∣z+1∣=z+2(1+i)

(x+1)

2

+y

2

=(x+iy)+2(1+i)

(x+1)

2

+y

2

+0i=(x+2)+(y+2)i

On equating real and imaginary parts, we get,

(x+1)

2

+y

2

=(x+2) and y=−2

y

2

=2x+3 and y=−2

x=

2

1

and y=−2

Hence,

z=

2

1

−2i.

hope it helps :)

Answer 2

$\text { Calculate the absolute value: }\left\{\begin{array}{l}\left|z^{2}+1\right|=1 \\z^{2}+1=1\end{array}\right.$

$\text { Rearrange unknown terms to the left side of the equation: }\left\{\begin{array}{l}\left|z^{2}+1\right|=1 \\z^{2}=1-1\end{array}\right.$

$\text { The sum of two opposites equals } 0: z^{2}=1-1$

$\text { Calculate the absolute value: }\left\{\begin{array}{l}z^{2}+1=1 \\z^{2}=0\end{array}\right.$

$\text { Rearrange unknown terms to the left side of the equation: }\left\{\begin{array}{l}z^{2}=1-1 \\z^{2}=0\end{array}\right.$

$\text { The sum of two opposites equals } 0: z^{2}=1-1$

The answer infinitely\ many\ solutions