+226
Let z and w be complex numbers satisfying \(|z| = 5\), \(|w| = 2\) and \(z\overline{w} = 6+8i\). Find in the numbers \(|z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2\)
+118667 I really do not know what I am doing here. (I have forgotten it)
However.
\(|z|=5\;\;\quad so\;\; \qquad z=5e^{\theta i}\\ |w|=2\;\;\quad so\;\; \qquad w=2e^{\alpha i}\\ |\bar w|=2\;\;\quad so\;\; \qquad \bar w=2e^{-\alpha i}\\~\\ zw=5e^{\theta i}*2e^{\alpha i}=10e^{(\theta+\alpha)i}\\ |zw|^2=100\\ \left| \frac{z}{w} \right|^2=\frac{25}{4}=6.25\\~\\ z\bar w=5e^{\theta i}*2e^{-\alpha i}=10e^{(\theta-\alpha)i}=6+8i\\ \theta-\alpha=atan(\frac{8}{6})=atan(1.\dot3)\\ z\bar w=6+8i=10e^{(atan(1.\dot 3))i} \)
I think that is right as far as it goes but it does not answer all your questions.
LaTex:
|z|=5\;\;\quad so\;\; \qquad z=5e^{\theta i}\\
|w|=2\;\;\quad so\;\; \qquad w=2e^{\alpha i}\\
|\bar w|=2\;\;\quad so\;\; \qquad \bar w=2e^{-\alpha i}\\~\\
zw=5e^{\theta i}*2e^{\alpha i}=10e^{(\theta+\alpha)i}\\
|zw|^2=100\\
\left| \frac{z}{w} \right|^2=\frac{25}{4}=6.25\\~\\
z\bar w=5e^{\theta i}*2e^{-\alpha i}=10e^{(\theta-\alpha)i}=6+8i\\
\theta-\alpha=atan(\frac{8}{6})=atan(1.\dot3)\\
z\bar w=6+8i=10e^{(atan(1.\dot 3))i}
