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Let z and w be complex numbers such that |z| = 1, |w| = 7, and |5z + w| = 11.  Find |3z + 4w|.

 Jul 29, 2022
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We can use the fact that for any complex number a, (a)(cong(a)) = |a|^2.

 

\(|5z + w|^2 = (5z + w)(5\overline{z} + \overline{w}) = 121 \)

\(25|z|^2 + |w|^2 + 5(\overline{z}w + \overline{w}z) = 121\)

\(5(\overline{z}w + \overline{w}z) = 47\)

 

\(|3z + 4w|^2 = (3z + 4w)(3\overline{z} + 4\overline{w}) \)

\(=793 + 12(\overline{w}z + \overline{z}w)\)

\(=793 + 564/5 \)

 

Take the square root of that value and you will the answer.

 Jul 31, 2022

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