1. ## [SOLVED] Transformation

Consider the transformation
z $\displaystyle \mapsto$ w = −$\displaystyle 5z^2+3$ .

(a) Find the image of the line $\displaystyle Re(z) = c$ (for c a non-zero real constant) under the
transformation, using as usual $\displaystyle z = x + iy$ and $\displaystyle w = u + iv$.

(b) Find the equation of the line $\displaystyle Im(z) = k$ (for k a real constant).

Help please explain how to solve this.

2. Originally Posted by ronaldo_07
Consider the transformation
z $\displaystyle \mapsto$ w = −$\displaystyle 5z^2+3$ .

(a) Find the image of the line $\displaystyle Re(z) = c$ (for c a non-zero real constant) under the
transformation, using as usual $\displaystyle z = x + iy$ and $\displaystyle w = u + iv$.
The equation $\displaystyle w=3-5z^2$ tells you that $\displaystyle u+iv = 3 - 5(x+iy)^2 = 5y^2 - 10 ixy + (3-5x^2)$. Put x=c and equate real and imaginary parts to get $\displaystyle u = 5y^2 + (3-5c^2)$ and $\displaystyle v = -10cy$. Solve the second of these for y and substitute that into the first: $\displaystyle u = \frac{v^2}{20c^2} + 3-5c^2$. You will recognise that as the equation of a parabola.

Originally Posted by ronaldo_07
(b) Find the equation of the line $\displaystyle Im(z) = k$ (for k a real constant).

Help please explain how to solve this.
I would have said that the equation of the line $\displaystyle \text{Im}(z) = k$ is $\displaystyle \text{Im}(z) = k$ (or equivalently y=k). But perhaps the question is meant to ask for the image of that line under the z→w transformation. If so, do it the same way as part (a).