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Let $$z$$ and $$w$$ be complex numbers satisfying $$|z| = 5, |w| = 2,$$ and $$z\overline{w} = 6+8i.$$ Then enter in the numbers $$|z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2$$, in the order listed above.

Thanks!!

Apr 12, 2019

#1
+30655
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$$\text{Let } z=a+bi\text{ and }w=c+di\\\text{Then }|z|=\sqrt{a^2+b^2}\text{, }|w|=\sqrt{c^2+d^2} \text{ and }z\bar w=(a+bi)(c-di)=ac+bd+(bc-ad)i$$

$$\text{Hence }a^2+b^2=5^2,ac+bd=6,c^2+d^2=2^2,bc-ad=8$$

(1)

$$|z+w|=|a+c+(b+d)i|=\sqrt{a^2+c^2+2ac+b^2+d^2+2bd}=\sqrt{a^2+b^2+c^2+d^2+2(ac+bd)}\\=\sqrt{5^2+2^2+2\times6}=\sqrt{41}\\\text{So }|z+w|^2=41$$

See if you can do the rest using a similar approach.

Apr 12, 2019
edited by Alan  Apr 12, 2019
#2
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I am able to do all of them except the third one. Can you help me on that as well?

Thanks!!

Apr 13, 2019
#3
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anyone?

Apr 13, 2019
#4
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Not sure which you meant by the third (including or excluding the one I did!), so:

Apr 14, 2019