+0

# complex numbers

0
133
1

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

Jul 29, 2022

#1
+285
0

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