9. Given that point $P$ is a moving point on the line $x+2 y=4$, and through point $P$ two tangents are drawn to the ellipse $x^{2}+4 y^{2}=4$, with the points of tangency being $A$ and $B$, when point $P$ moves, the line $AB$ passes through a fixed point with coordinates
Answer $\left(1, \frac{1}{2}\right)$.
Analysis Let $P\left(x_{0}, y_{0}\right)$, then $x_{0}+2 y_{0}=4$, i.e., $y_{0}=\frac{1}{2}\left(4-x_{0}\right)$, and the chord equation of the tangent points $A B$ is $x_{0} x+4 y_{0} y=4$, i.e.,
$$
x_{0} x+4 y \cdot \frac{1}{2}\left(4-x_{0}\right)=4, \quad(x-2 y) x_{0}+(8 y-4)=0
$$

Thus $x-2 y=0,8 y-4=0$, solving gives $x=1, y=\frac{1}{2}$, i.e., the coordinates of the fixed point through which the line $A B$ passes are $\left(1, \frac{1}{2}\right)$, hence fill in $\left(1, \frac{1}{2}\right)$.