# Complex numbers - finding minimum value

For all complex numbers $z_1,z_2$ satisfying $|z_1|=12$ and $|z_2-3-4i|=5$ , find the minimum value of $|z_1-z_2|$

Can we go like this :

Let $z_1 = x +iy$ therefore $|z_1| = \sqrt{x_1^2+y_1^2}$ and $z_2 = x_2+ iy_2$

$|z_2-3-4i| = \sqrt{(x_2-3)^2+ (y_2-4)^2}$ so $x_2^2-6x_2+9+y_2^2-8y_2+16 =25$ ( squaring both sides) .. Please guide if it is correct..

• is that $z_1$ in $|z-1-z_2|$ – Maesumi Apr 11 '13 at 1:27

Another hint: Hence, the maximum is $$22$$ and the minimum is $$2$$ but you should prove this.

• Don't you mean $2$? We are looking for the minimum distance. – Ross Millikan Apr 12 '13 at 3:39

An algebraic method

$$|z_1 - z_2 | \geq |z_1| - |z_2| \\ |z_2-(3+4i)| \geq |z_2| - 5 \\ |z_2|\leq 10\\ |z_1 - z_2 | \geq 12-10=2$$

Hint: Draw the sets $\mathcal{A} = \{z_1: |z_1| = 12\}$ and $\mathcal{B} = \{z_2: |z_2-3-4i| = 5\}$ in the complex plane (both will be circles). Now draw the set $\mathcal{B}' = \{1+z_2:|z_2-3-4i| = 5\}$, which will be "$\mathcal{B}$ shifted one unit to the right." You would like to find the two closest points of $\mathcal{A}$ and $\mathcal{B}'$. The answer will be obvious as soon as you draw these circles.

• nice solution Lord Soth..... – juantheron Dec 15 '13 at 11:11
• @juantheron, I am new to complex numbers and trying to understand why $1+z_{2}$ is plotted in this scenario? – Sunil Feb 1 '17 at 6:23