5. Find all values of the parameter $a$, for which the area of the quadrilateral bounded on the coordinate plane by the x-axis and the lines $x=a+5, x=a+1$, $y=(a-3)x$, is 28.
Solution. The lines $x=a+5, x=a+1$ are vertical and the distance between them is 4 for any value of the parameter $a$.

![](https://cdn.mathpix.com/cropped/2024_05_06_c6cf128dca72ce13eb74g-2.jpg?height=960&width=1251&top_left_y=1262&top_left_x=194)

The line $y=(a-3) x$ passes through the origin. These lines and the x-axis bound a quadrilateral on the coordinate plane if either $a+1>0$, or $a+5 - 1 }
\end{array} }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ | a ^ { 2 } - 9 | = 7 } \\
{ [ \begin{array} { l } 
{ a  - 1 }
\end{array} }
\end{array} }
\end{array} \Leftrightarrow \left[\begin{array}{l}
a=4 \\
a=\sqrt{2}
\end{array}\right.\right.\right.
$$

Answer: $\sqrt{2} ; 4$.