Given that $\displaystyle z=w^2$ where $\displaystyle z=x+yi$ and $\displaystyle w=u+vi$, prove that:

a) ifuvaries butvis constant, then the locus of $\displaystyle P(x,y)$ is a parabola,

b) ifvvaries anduis constant, then the locus of P is again a parabola.

Show that these two parabolas have the same axis and focus.

From the relation:

$\displaystyle x+yi=u^2-v^2+2uvi$

$\displaystyle x=u^2-v^2$

$\displaystyle y=2uv$

Which is the parametric equation for a parabola.

Whenuvaries, the parabola has focus $\displaystyle (v,0)$ and axis $\displaystyle y=0$.

Whenvvaries, the parabola has focus $\displaystyle (u,0)$ and axis $\displaystyle y=0$.

The parabolas face in opposite directions, and both cross the y-axis, assuming that the constants are greater than 0. My problem is how do I show that they have the same focus. They do have the same axis.

Thanks!