Determine the set of points $M$ of affix of $z\in\mathbb{C}$ such that there exists at least one real $t$ satisfying $z^2=t(t-i)$
My attempt:
We look for the form $z=x+yi$ and we want there is a real $t$ such that $\ x^2-y^2+2xyi=t^2-ti\ $ i.e $\left\{\matrix{x^2+y^2=t^2\cr 2xy=-t}\right.$
The second equation leaves no choice: if such a $t$ exists is necessarily $t=-2xy$ and, for the first equation is verified, it is necessary and sufficient that $x^2+y^2=4x^2y^2$, that is, $y=\dfrac{\pm x}{\sqrt{4x^2-1}}$ with $|x|>\frac{1}{2}$ which, to my knowledge, is the equation ... nothing special ...
- Am i right ?
- Is there other ways to solve it
- how can i explain solution to student just read the complexe number