Question

Let $z$ and $w$ be complex numbers satisfying $|z| = 5, |w| = 2,$ and $z\overline{w} = 6+8i.$ Then enter in the numbers \[|z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2 \]below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter in a question mark. Don't even know where to start.

I got 49 for the first one, 100 for the second one, 9 for the 3rd one, and 6.25 for the 4th one. But it was wrong, so I don't know how to do this question.

Answer 1

1st entry: 41

3rd entry: 17

Explanation:

The 1st and 3rd entry are incorrect.

We will use the following for the 1st and 3rd:

`|z+w|=|z|^2+|w|^2+2R(z\overline{w})`

`|z-w|=|z|^2+|w|^2-2R(z\overline{w})`

where
`R(z\overline{w})` means 'real part of
`z\overline{w}`'.

Let's do part 1 now)

25+4+2(6)

29+12

41

Let's do part 3 now)

25+4-2(6)

29-12

17