Problem 9.7. On the coordinate plane, points $O(0 ; 0), A(5 ; 0), B(0 ; 4)$ are marked. The line $y=k x+b$ is such that for any point $M$ on this line, the area of the quadrilateral $A O B M$ is 20. What is $k$?
Answer: $-0.8$.

![](https://cdn.mathpix.com/cropped/2024_05_06_ce887724b4f1896529bag-5.jpg?height=362&width=420&top_left_y=1139&top_left_x=517)

Fig. 1: to the solution of problem 9.7

Solution. For any point $M$ from the condition of the problem, the following is true

$$
S_{A B M}=S_{O A M B}-S_{A B O}=20-10=10
$$

On the other hand,

$$
S_{A B M}=\frac{A B \cdot h}{2}
$$

where $h$ is the height from point $M$ to the line $A B$ (Fig. 1). But this means that $h$ does not depend on the position of point $M$, which is possible only if the line $y=k x+b$ is parallel to the line $A B$. Then their slopes coincide.

And it is not difficult to obtain the equation of the line $A B$: $y=-0.8 x+4$.