Problem 9. Form the equation of the plane $\alpha$, relative to which the points $D(-5 ; 6 ;-1)$ and $B(3 ; 4 ; 1)$ are symmetric.
Solution. The plane $\alpha$ intersects the segment $DB$ at point $P(-1; 5; 0)$, which is the midpoint of segment $DB$. Let $M(x; y; z)$ be an arbitrary point on the plane $\alpha$. Then $\overline{DB} \cdot \overline{PM}=0$, i.e., $(8; -2; 2) \times (x+1; y-5; z)=0$, or $8(x+1)-2(y-5)+2z=0$.

Thus, the equation of the plane $\alpha$ is: $4x - y + z + 9 = 0$.

![](https://cdn.mathpix.com/cropped/2024_05_21_8debcc8c8285c4d79a3dg-125.jpg?height=326&width=508&top_left_y=268&top_left_x=360)

Fig. 2

![](https://cdn.mathpix.com/cropped/2024_05_21_8debcc8c8285c4d79a3dg-125.jpg?height=326&width=363&top_left_y=268&top_left_x=958)

![](https://cdn.mathpix.com/cropped/2024_05_21_8debcc8c8285c4d79a3dg-125.jpg?height=311&width=257&top_left_y=256&top_left_x=1456)

Fig. 3