|2(a-iy)+i|=4

Lösung

∣ 2 (a - i y) + i ∣ = 4

y = x + \frac{- 2 a + - 4 x ^{2} + 4 x + 15}{2} i, y = x - \frac{2 a + - 4 x ^{2} + 4 x + 15}{2} i; x \in R

Schritte zur Lösung

∣ 2 (a - i y) + i ∣ = 4

Ersetze

y = x + v i

∣ 2 (a - i (x + v i)) + i ∣ = 4

Schreibe

∣ 2 (a - i (x + v i)) + i ∣

um:

4 a^{2} + 8 a v + 4 v^{2} + 4 x^{2} - 4 x + 1

4 a^{2} + 8 a v + 4 v^{2} + 4 x^{2} - 4 x + 1 = 4

Löse nach

v : v = \frac{- 2 a + - 4 x ^{2} + 4 x + 15}{2}, v = - \frac{2 a + - 4 x ^{2} + 4 x + 15}{2}

Setze in

y = x + v i

ein

y = x + \frac{- 2 a + - 4 x ^{2} + 4 x + 15}{2} i, y = x - \frac{2 a + - 4 x ^{2} + 4 x + 15}{2} i; x \in R

x^{2} = \frac{- 1 + i 3}{2}

x + 3 i = (1 + y i) (2 - 3 i)

x^{2} + (1 - 2 i) x - 2 i = 0

z^{2} - 3 + 2 i = 0, z

zz = 13 - 13 i