The mapping of curve $y = x^2$ by linear transformation $w = 2iz + (1-i)$ is....
I have looked at my textbook to solve this problem and also searched some materials in internet, but I don't found any solution for this. This is a MCQ and the choice leads to $u$ and $v$ variables to use.
The answer is the parabola $(1+v)^{2}=2(1-u)$. The transformed curve consists of the points $(1-2x^{2})+i(2x-1)$. Write $u=1-2x^{2}$ and $v=2x-1$. Use the second equation to solve for $x$. You get $x=\frac {v+1} 2$. Substitute this into the first equation to get a formula satisfied by $u$ and $v$. A little simplification gives $(1+v)^{2}=2(1-u)$.