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Let z and w be complex numbers satisfying |z| = 5, |w| = 2, and z(overline{w}) = 6+8i.

Find the value of |z+w|^2, |zw|^2, |z-w|^2,  |frac{z}{w}|^2 .

If any of these cannot be uniquely determined from the information given, write not possible.

May 10, 2019

#1
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Let $$z$$ and $$w$$ be complex numbers satisfying $$|z| = 5$$, $$|w| = 2$$, and $$z\overline{w} = 6+8i$$

Find the value of $$|z+w|^2$$, $$|zw|^2$$, $$|z-w|^2$$,  $$\left|\dfrac{z}{w}\right|^2$$ .

$$\begin{array}{|rcll|} \hline && \mathbf{|z+w|^2} \\ &=& |z|^2+|w|^2+2*\Re{(z\overline{w})} \\ &=& 5^2+2^2+2*6 \quad | \quad \Re{(z\overline{w})} = \Re{(6+8i)} = 6 \\ &=& \mathbf{41} \\ \hline && \mathbf{|zw|^2} \\ &=&|z|^2|w|^2 \\ &=& 5^2*2^2 \\ &=& 25*4 \\ &=& \mathbf{100} \\ \hline && \mathbf{|z-w|^2} \\ &=& |z|^2+|w|^2-2*\Re{(z\overline{w})} \\ &=& 5^2+2^2-2*6 \quad | \quad \Re{(z\overline{w})} = \Re{(6+8i)} = 6 \\ &=& \mathbf{17} \\ \hline && \mathbf{\left|\dfrac{z}{w}\right|^2} \\ &=& \dfrac{|z|^2}{|w|^2} \\ &=& \dfrac{5^2}{2^2} \\ &=& \dfrac{25}{4} \\ &=& \mathbf{6.25} \\ \hline \end{array}$$

May 10, 2019