What is the answer to |a-1+2i|=3 ?

The answer to |a-1+2i|=3 is a=(1+sqrt(-y^2-4y+5))+yi,a=(1-sqrt(-y^2-4y+5))+yi;\quad y\in \mathbb{R}

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|a-1+2i|=3

∣ a - 1 + 2 i ∣ = 3

a = (1 + - y^{2} - 4 y + 5) + y i, a = (1 - - y^{2} - 4 y + 5) + y i; y \in R

Solution steps

∣ a - 1 + 2 i ∣ = 3

Substitute

a = x + y i

∣ x + y i - 1 + 2 i ∣ = 3

Expand

∣ x + y i - 1 + 2 i ∣ : x^{2} - 2 x + y^{2} + 4 y + 5

x^{2} - 2 x + y^{2} + 4 y + 5 = 3

Solve for

x : x = 1 + - y^{2} - 4 y + 5, x = 1 - - y^{2} - 4 y + 5

Substitute back

a = x + y i

a = (1 + - y^{2} - 4 y + 5) + y i, a = (1 - - y^{2} - 4 y + 5) + y i; y \in R

z - (1 + i) w = 2

6 + 7 i = x + y i

\frac{4 i}{- 6 + 4 i} = \frac{a}{b}

\frac{15}{z - 1} = - 6 - 3 i

(x - 2) i = y - 3 i

What is the answer to |a-1+2i|=3 ?
The answer to |a-1+2i|=3 is a=(1+sqrt(-y^2-4y+5))+yi,a=(1-sqrt(-y^2-4y+5))+yi;\quad y\in \mathbb{R}