Consider the transformation
z w = − .
(a) Find the image of the line (for c a non-zero real constant) under the
transformation, using as usual and .
(b) Find the equation of the line (for k a real constant).
Help please explain how to solve this.
Consider the transformation
z w = − .
(a) Find the image of the line (for c a non-zero real constant) under the
transformation, using as usual and .
(b) Find the equation of the line (for k a real constant).
Help please explain how to solve this.
The equation tells you that . Put x=c and equate real and imaginary parts to get and . Solve the second of these for y and substitute that into the first: . You will recognise that as the equation of a parabola.
I would have said that the equation of the line is (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).