What is the answer to |a-i|=sqrt(37) ?

The answer to |a-i|=sqrt(37) is a=x+(1+sqrt(-x^2+37))i,a=x+(1-sqrt(-x^2+37))i;\quad x\in \mathbb{R}

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|a-i|=sqrt(37)

∣ a - i ∣ = 37

a = x + (1 + - x^{2} + 37) i, a = x + (1 - - x^{2} + 37) i; x \in R

Solution steps

∣ a - i ∣ = 37

Substitute

a = x + y i

∣ x + y i - i ∣ = 37

Expand

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

x^{2} + y^{2} - 2 y + 1 = 37

Solve for

y : y = 1 + - x^{2} + 37, y = 1 - - x^{2} + 37

Substitute back

a = x + y i

a = x + (1 + - x^{2} + 37) i, a = x + (1 - - x^{2} + 37) i; x \in R

x^{2} - 2 i x - 2 = 0

(x + y i) + 4 (1 - i) = 5 - 7 i

z^{2} - (5 + 4 i) z + (3 - 5 i) = 0

x^{2} + 3 x + 2 - i = 0

x^{2} - 3 + 4 i = 0

What is the answer to |a-i|=sqrt(37) ?
The answer to |a-i|=sqrt(37) is a=x+(1+sqrt(-x^2+37))i,a=x+(1-sqrt(-x^2+37))i;\quad x\in \mathbb{R}