Complex numbers problem

[scermat]

Hi there,

Can anyone help me with this question please?

i) If the real part of (z+2)/(z+2i) is equal to 1, show that the point z lies on a straight line in the Argand diagram and find the equation of this line.

ii)Hence find the point a on this line such that |a|=√2

iii)Find also the quadratic equation with real coefficients which has a as one of the roots.

I managed to arrive to an answer for i) but I'd appreciate if someone can confirm this please.

i) y=2-x

Thanks for any help; I appreciate it

 

[pka]

RULE #1: When working with a fraction in the complex numbers ALWAYS simplify first.
\(\displaystyle \L
\frac{1}{z} = \frac{{\bar z}}{{\left| z \right|^2 }}\quad \Rightarrow \quad \frac{{z + 2}}{{z + 2i}} = \frac{{\left( {z + 2} \right)\left( {\bar z - 2i} \right)}}{{\left| {z + 2i} \right|^2 }} = \frac{{z\bar z - 2zi + 2\bar z - 4i}}{{\left| {z + 2i} \right|^2 }}\)

Now some ground work.
Because \(\displaystyle {\left| {z + 2i} \right|^2 }\) is a real number we can do the following:
\(\displaystyle \begin{array}{l}
Re(z\bar z - 2zi + 2\bar z - 4i) = x^2 + y^2 + 2y + 2x \\
Re\left( {\left| {z + 2i} \right|^2 } \right) = x^2 + (y + 2)^2 \\
\end{array}\)

Thus \(\displaystyle \L
Re\left( {\frac{{z + 2}}{{z + 2i}}} \right) = \frac{{x^2 + y^2 + 2y + 2x}}{{x^2 + (y + 2)^2 }}\).

 

[scermat]

thanks a lot!