What is the answer to |Z+5i|=iZ ?

The answer to |Z+5i|=iZ is Z=-5/2 i

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|Z+5i|=iZ

∣ Z + 5 i ∣ = i Z

Z = - \frac{5}{2} i

Solution steps

∣ Z + 5 i ∣ = i Z

Substitute

Z = x + y i

∣ x + y i + 5 i ∣ = i (x + y i)

Expand

x^{2} + y^{2} + 10 y + 25 = - y + i x

Rewrite

x^{2} + y^{2} + 10 y + 25

in standard complex form:

x^{2} + y^{2} + 10 y + 25 + 0 i

x^{2} + y^{2} + 10 y + 25 + 0 i = - y + i x

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[x^{2} + y^{2} + 10 y + 25 = - y 0 = x]

[x^{2} + y^{2} + 10 y + 25 = - y 0 = x] : (x = 0, y = - \frac{5}{2})

Substitute back

Z = x + y i

Z = - \frac{5}{2} i

(2 a + 1) + (2 b + 3) i = 5 + 12 i

0.022 i = \frac{1}{x \cdot i}

3 x + 4 y + (x + 2 y) i = 5 + 3 i

r (t) = (5 + t) i + t j

a, z = \frac{1 - ai}{1 + ai}