20. Among the complex numbers $z$, find the numbers having the smallest and largest arguments, if $|z-5-5i|=5$.
20. (Fig. 20) The desired number has the form \( z = x + yi \). Then \( |x \div yi - 5 - 5i| = 5 \) or \( \sqrt{(x-5)^2 + (y-5)^2} = 5 \). The smallest value of \( \varphi = 0 \). The largest \( \varphi = 90^\circ \). Answer: \( z_1 = 5, \quad z_2 = 5i \).

![](https://cdn.mathpix.com/cropped/2024_05_21_ec9ddf54a7c86775293eg-092.jpg?height=286&width=916&top_left_y=1453&top_left_x=567)

Fig. 18

![](https://cdn.mathpix.com/cropped/2024_05_21_ec9ddf54a7c86775293eg-092.jpg?height=243&width=757&top_left_y=2009&top_left_x=364)

Fig. 19

![](https://cdn.mathpix.com/cropped/2024_05_21_ec9ddf54a7c86775293eg-092.jpg?height=385&width=466&top_left_y=1869&top_left_x=1229)

Fig. 20

6*

91

![](https://cdn.mathpix.com/cropped/2024_05_21_ec9ddf54a7c86775293eg-093.jpg?height=617&width=491&top_left_y=248&top_left_x=360)

Fig. 21

![](https://cdn.mathpix.com/cropped/2024_05_21_ec9ddf54a7c86775293eg-093.jpg?height=548&width=443&top_left_y=317&top_left_x=1252)

Fig. 22