Let z and w be complex numbers such that |z| = 1, |w| = 7, and |5z + w| = 11. Find |3z + 4w|.
+289 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.
