So I am stuck on the following problem from "Edexcel AS and A Level Modular Mathematics FP$1$":
$z=\frac {1+7i}{4+3i}$
a Find the modulus and argument of $z$.
b Write down the modulus and argument of $z^*$.
In an Argand diagram, the points $A$ and $B$ represent $1+7i$ and $4+3i$ respectively and $O$ is the origin. The quadrilateral $OABC$ is a parallelogram.
c Find the complex number represented by the point $C$.
d Calculate the area of the parallelogram.
For a, I have found that $|z|=\sqrt 2$ and $arg (z)=\frac {\pi}4$.
For b, I have found that $|z^*|=\sqrt 2$ and $arg (z)=-\frac {\pi}4$.
But now I find myself in a quandary.
For c, I had found that $C$ could be represented with $5+10i.$
a and b are correct, but it shows $3-4i$ as the answer. Upon further thought, I realized that there were $3$ possible answers for c: $5+10i, 3-4i$ and $-3+4i$, as shown by:
Is any answer correct in particular or do I simply choose any one and proceed to calculate the area?
Hint:
All three solutions yield a paralelogram, however only one solution has the corners of the paralelogram in the correct order!