# Thread: Image of line in complex plane

1. ## Image of line in complex plane

The question:

Find the image of the line x - y = 2 in the complex plane under the mapping w = iz - 1.

My solution:

$\displaystyle z = \frac{w + 1}{i}$

Let z = x + iy, w = a + ib

$\displaystyle \frac{(a + 1) + ib}{i} . \frac{-i}{-i}$

b - i(a + 1)

Equating real and imaginary parts,

x = b
y = (a + 1)

Sub into equation of the line:

b + (a + 1) = 2
b = 1 - a

Is this correct? Thanks.

2. Originally Posted by Glitch
The question:

Find the image of the line x - y = 2 in the complex plane under the mapping w = iz - 1.

My solution:

$\displaystyle z = \frac{w + 1}{i}$

Let z = x + iy, w = a + ib

$\displaystyle \frac{(a + 1) + ib}{i} . \frac{-i}{-i}$

b - i(a + 1)

Equating real and imaginary parts,

x = b
y = (a + 1)

Sub into equation of the line:

b + (a + 1) = 2
b = 1 - a

Is this correct? Thanks.
Yes your answer is correct also note that since the map is a Linear Fractional Transformation we know that it maps lines and circles to lines and circles.

Since the transformation is not bounded it must map this line to another line so we really only need to transform two points and find the equation of the line.

$\displaystyle w(x,y)=iz-1=-(y+1)+ai$ this gives

$\displaystyle w(2,0)=(-1,2) \quad w(0,-2)=(1,0)$

using these two point gives

$\displaystyle y=-x+1\iff x+y=1$ which matches your answer!