# Math Help - Complex numbers

1. ## Complex numbers

The complex numbers Z and W are represented, respectively, by points P(x, y) and Q(u,v) in Argand diagrams and

W=Z²

a) Obtain expressions for u and v in terms of x and y.

b) the point P moves along the curve with equation y²=2x²-1. Find the cartesian equation of the locus of Q.

I think i have done part a correctly ...

u + iv = (x+iy)²
u + iv = x² + 2iyx - y²

u = x² - y²

v = 2yx

I just have no idea how to do part B. Usually the equations dont have squared terms and the equations can be rearranged easily im just not sure how to do it when they are squared

any help??

thanks

2. Originally Posted by djmccabie
The complex numbers Z and W are represented, respectively, by points P(x, y) and Q(u,v) in Argand diagrams and

W=Z²

a) Obtain expressions for u and v in terms of x and y.

b) the point P moves along the curve with equation y²=2x²-1. Find the cartesian equation of the locus of Q.
Let $w=z^2$ be a complex function $w: \mathbb{C}\to \mathbb{C}$.
Notice that $w(x+iy) = (x+iy)^2 = (x^2 - y^2) + 2ixy$.

If $P$ is on the curve $y^2 = 2x^2 - 1$ it means $P = x+iy$ where $y^2 = 2x^2 - 1$. But then $w(P) = (x^2 - y^2) + 2ixy$. Therefore, $w(P) = (1 - x^2) + 2ixy$. Therefore, $w(P)$ is on the curve $2xy = 1-x^2$.