What is the answer to |z-3+4i|=5 ?

The answer to |z-3+4i|=5 is z=(3+sqrt(-y(y+8)+9))+yi,z=(3-sqrt(-y(y+8)+9))+yi;\quad y\in \mathbb{R}

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|z-3+4i|=5

∣ z - 3 + 4 i ∣ = 5

z = (3 + - y (y + 8) + 9) + y i, z = (3 - - y (y + 8) + 9) + y i; y \in R

Solution steps

∣ z - 3 + 4 i ∣ = 5

Substitute

z = x + y i

∣ x + y i - 3 + 4 i ∣ = 5

Expand

∣ x + y i - 3 + 4 i ∣ : x^{2} - 6 x + y^{2} + 8 y + 25

x^{2} - 6 x + y^{2} + 8 y + 25 = 5

Solve for

x : x = 3 + - y (y + 8) + 9, x = 3 - - y (y + 8) + 9

Substitute back

z = x + y i

z = (3 + - y (y + 8) + 9) + y i, z = (3 - - y (y + 8) + 9) + y i; y \in R

(3 - 2 i) (x + i) = 17 + i y

z^{2} + z \cdot i + 2 = 0

(a + 4 i) + (7 + bi) = 5 + 7 i

x - i y = - 2 + 3 i

\frac{( 1 + ai )}{5 + bi} = 1 + 5 i

What is the answer to |z-3+4i|=5 ?
The answer to |z-3+4i|=5 is z=(3+sqrt(-y(y+8)+9))+yi,z=(3-sqrt(-y(y+8)+9))+yi;\quad y\in \mathbb{R}