10.36 Find the real number $b$ from the condition that the points representing the complex numbers $3-5 i, 1-i$ and $-2+b i$ lie on the same straight line.
10.36 The numbers $3-5i, 1-i$ and $-2+bi$ correspond to the points $A(3, -5), B(1, -1)$ and $C(-2, b)$ on the plane (Fig. 10.10). We find the coordinates of the vectors $\overline{A B}$ and $\overline{A C}$; we have $\overline{A B}(-2, 4), \overline{A C}(-5, b+5)$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-332.jpg?height=328&width=375&top_left_y=482&top_left_x=782)

Fig. 10.10

Since these points lie on the same line, the equality $\overline{A C}=k \overline{A B}$ must hold,

i.e.

$$
\left\{\begin{array}{l}
-2k = -5 \\
4k = b + 5
\end{array}\right.
$$

From this system, we find $b=5$.

Answer: 5.