Let z be a complex number such that the conjugate of (z - 1), multiplied by (4 + 8i), equals 2. Find the value of the modulus of z squared.

Step-by-step solution

1. 1

   Decide why you cannot shortcut with moduli alone. Taking absolute values gives $|z-1|\cdot|4+8i|=2$, i.e. $|z-1|=\tfrac{2}{\sqrt{80}}$. That is the distance from $z$ to $1$, not $|z|$, so it does not determine $|z|^2$. We must actually recover $z$ itself.

2. 2

   Isolate the conjugate and rationalise. Divide both sides by $4+8i=4(1+2i)$ and multiply by the conjugate of the denominator:

   $\overline{(z-1)}=\frac{2}{4(1+2i)}=\frac{1}{2(1+2i)}\cdot\frac{1-2i}{1-2i}=\frac{1-2i}{2(1+4)}=\frac{1-2i}{10}.$

3. 3

   Conjugate both sides to undo the bar. Conjugation is its own inverse, and $\overline{\overline{w}}=w$, so

   $z-1=\overline{\left(\frac{1-2i}{10}\right)}=\frac{1+2i}{10}=\frac{1}{10}+\frac{1}{5}i.$

4. 4

   Solve for $z$. Adding $1$:

   $z=1+\frac{1}{10}+\frac{1}{5}i=\frac{11}{10}+\frac{1}{5}i.$

5. 5

   Apply $|z|^2=a^2+b^2$. With $a=\tfrac{11}{10}$ and $b=\tfrac15$:

   $|z|^2=\frac{121}{100}+\frac{1}{25}=\frac{121}{100}+\frac{4}{100}=\frac{125}{100}=\frac54.$

   Check: $\overline{(z-1)}(4+8i)=\left(\tfrac{1-2i}{10}\right)(4+8i)=\tfrac{4+8i-8i+16}{10}=2$, as required.