12.6. Determine the minimum value of the modulus of the complex number $z$, which satisfies the relation

$$
|z+12|+|z-5 i|=13
$$
## Solution:

If $P\left(a_{1}, b_{1}\right)$ is the image of the complex number

$z_{1}=a_{1}+b_{1} i$, and $Q\left(a_{2}, b_{2}\right)$ is the image of the complex number

$z_{2}=a_{2}+b_{2} i$, then $\left|z_{1}-z_{2}\right|=P Q$.

Let $M$ be the image of $z_{1}=-12$, $N$ be the image of $z_{2}=5 i$, and $A$ be the image of $z$. Then $|z+12|=A M$, $|z-5 i|=A N$, $13=M N$. Then $|z+12|+|z-5 i|=13 \Leftrightarrow A M+A N=$ $M N \Rightarrow A \in[M N]$. The minimum value of the length of the segment $O A, O A=|z|$, is the distance from point $O$ to the line $M N$ and is equal to $\frac{12 \cdot 5}{13}=\frac{60}{13}$.

![](https://cdn.mathpix.com/cropped/2024_06_06_bf7dcdb792e591bc55feg-1.jpg?height=458&width=696&top_left_y=1468&top_left_x=1338)