5. Four lines $l_{1}: x+3 y-15=0, l_{2}: k x-y-6=0, l_{3}: x+5 y=$ $0, l_{4}: y=0$ form a quadrilateral. For what value of $k$ does this quadrilateral have a circumscribed circle, and find the equation of this circumscribed circle.
5. Let the equation of the quadratic curve system passing through the 4 vertices of the quadrilateral be
$$
\begin{array}{l}
(x+3 y-15)(x+5 y)+\lambda(k x-y-6) y=0, \\
\text { from } \left\{\begin{array} { l } 
{ 1 5 - \lambda = 1 } \\
{ 8 + k \lambda = 0 }
\end{array} \text { we get } \left\{\begin{array}{l}
\lambda=14, \\
k=-\frac{4}{7}
\end{array} \text { Therefore, the equation of the required circle is } x^{2}+y^{2}-15 x-159 y=0\right.\right. \text {. }
\end{array}
$$