4. Given the ellipse $a^{2} x^{2}+b^{2} y^{2}=5 c^{2}$ intersects with the line $a x+$ $b y=c$ at points $A$ and $B$. If $a$, $b$, and $c$ form a geometric sequence, then $\frac{b}{a}\left(k_{O A}+k_{O B}\right)=$ $\qquad$.
4. $-\frac{5}{2}$.

Let the common ratio be $q$. Combining the equations of the ellipse and the line, we get
$$
\begin{array}{l}
\left\{\begin{array}{l}
x^{2}+q^{2} y^{2}=5 q^{4}, \\
x+q y=q^{2}
\end{array}\right. \\
\Rightarrow(x, y)=\left(-q^{2}, 2 q\right),\left(2 q^{2},-q\right) . \\
\text { Then the original expression }=q\left(-\frac{2}{q}-\frac{1}{2 q}\right)=-\frac{5}{2} .
\end{array}
$$