# Transformations in the Compex Plane

• Nov 8th 2009, 10:42 AM
dbatten
Transformations in the Compex Plane
For the transformation w=1/(z-2), where z=x+iy and w=u+iv,
show that the straight line 2x+y=5 is transformed to a circle with centre (1,-0.5) and radius 0.5sqrt5.

I can get to the stage where I equate the real and imaginary parts, but I can't see how to use the information about the original image, i.e the straight line 2x+y=5.

Any help would be greatly appreciated, thanks.
• Nov 9th 2009, 09:19 AM
Opalg
Quote:

Originally Posted by dbatten
For the transformation w=1/(z-2), where z=x+iy and w=u+iv,
show that the straight line 2x+y=5 is transformed to a circle with centre (1,-0.5) and radius 0.5sqrt5.

I can get to the stage where I equate the real and imaginary parts, but I can't see how to use the information about the original image, i.e the straight line 2x+y=5.

Given w = 1/(z–2), solve that for z to get $\displaystyle z = \frac1w+2 = \frac{u-iv}{u^2+v^2}+2$. Take real and imaginary parts: $\displaystyle x = \frac u{u^2+v^2}+2$, $\displaystyle y = \frac{-v}{u^2+v^2}.$ Now substitute those values of x and y into the equation 2x+y=5, and you get the equation for u and v.