Question

Given $|z_1|=1$, $|z_2-2|=3$ and $|z_3-5|=6$, find the maximum value of $|2z_1-3z_2-4z_3|$, where $z_1, z_2, z_3$ are complex numbers.

Answer 1

61

Answer 2

Let the given constraints be:

$|z_1|=1$ $|z_2-2|=3$ $|z_3-5|=6$

We want to find the maximum value of $|2z_1-3z_2-4z_3|$. We can rewrite the expression by introducing the centers of the circles from the constraints:

$|2z_1-3z_2-4z_3| = |2z_1 - 3(z_2-2+2) - 4(z_3-5+5)|$ $= |2z_1 - 3(z_2-2) - 6 - 4(z_3-5) - 20|$ $= |2z_1 - 3(z_2-2) - 4(z_3-5) - 26|$

Let $w_1 = z_1$, $w_2 = z_2-2$, and $w_3 = z_3-5$. The given constraints become $|w_1|=1$, $|w_2|=3$, and $|w_3|=6$. We want to maximize $|2w_1 - 3w_2 - 4w_3 - 26|$.

Using the triangle inequality, $|a+b+c+d| \le |a|+|b|+|c|+|d|$:

$|2w_1 - 3w_2 - 4w_3 - 26| \le |2w_1| + |-3w_2| + |-4w_3| + |-26|$ $|2w_1 - 3w_2 - 4w_3 - 26| \le |2||w_1| + |-3||w_2| + |-4||w_3| + |-26|$ $|2w_1 - 3w_2 - 4w_3 - 26| \le 2|w_1| + 3|w_2| + 4|w_3| + 26$

Substitute the values of $|w_1|$, $|w_2|$, and $|w_3|$:

$|2w_1 - 3w_2 - 4w_3 - 26| \le 2(1) + 3(3) + 4(6) + 26$ $|2w_1 - 3w_2 - 4w_3 - 26| \le 2 + 9 + 24 + 26$ $|2w_1 - 3w_2 - 4w_3 - 26| \le 61$

The maximum value is thus at most 61. For the equality to hold in the triangle inequality $|a+b+c+d| \le |a|+|b|+|c|+|d|$, the complex numbers $a, b, c, d$ must be collinear and have the same argument. In our case, the terms are $a = 2w_1$, $b = -3w_2$, $c = -4w_3$, and $d = -26$. Since $d = -26$ is a negative real number, its argument is $\arg(-26) = \pi$.

For equality, we require $\arg(2w_1) = \pi$, $\arg(-3w_2) = \pi$, $\arg(-4w_3) = \pi$, and $\arg(-26) = \pi$.

1. $\arg(2w_1) = \pi \implies \arg(w_1) = \pi$. Since $|w_1|=1$, $w_1 = 1 \cdot e^{i\pi} = -1$.
2. $\arg(-3w_2) = \pi \implies \arg(-3) + \arg(w_2) = \pi \implies \pi + \arg(w_2) = \pi \implies \arg(w_2) = 0$. Since $|w_2|=3$, $w_2 = 3 \cdot e^{i0} = 3$.
3. $\arg(-4w_3) = \pi \implies \arg(-4) + \arg(w_3) = \pi \implies \pi + \arg(w_3) = \pi \implies \arg(w_3) = 0$. Since $|w_3|=6$, $w_3 = 6 \cdot e^{i0} = 6$.

Now we find the corresponding values of $z_1, z_2, z_3$:

$w_1 = z_1 = -1$. This satisfies $|z_1|=1$. $w_2 = z_2-2 = 3 \implies z_2 = 2+3 = 5$. This satisfies $|z_2-2|=|5-2|=3$. $w_3 = z_3-5 = 6 \implies z_3 = 5+6 = 11$. This satisfies $|z_3-5|=|11-5|=6$.

Since we found values of $z_1, z_2, z_3$ that satisfy the given constraints and make the triangle inequality an equality, the maximum value is indeed 61.

Let's verify the value of the expression for these $z_i$:

$|2z_1 - 3z_2 - 4z_3| = |2(-1) - 3(5) - 4(11)| = |-2 - 15 - 44| = |-61| = 61$.

Therefore, the maximum value is 61.