3. Find the equation of the parabola that is tangent to the $x$-axis and $y$-axis at points $(1,0)$ and $(0,2)$, respectively, and find the axis of symmetry and the coordinates of the vertex of the parabola.
(The origin $(0,0)$ is related to the chord of tangents of the parabola, which is the line $l: 2 x+y-2=0$ passing through the tangent points $(1,0)$ and $(0,2)$. It is easy to find that the equation of the parabola is
$$
(2 x-y)^{2}-4(2 x+y)+4=0 \text {. }
$$

Therefore, the axis of symmetry of the parabola is $10 x-5 y-6=0$, and the vertex is $\left(\frac{16}{25}, \frac{2}{25}\right)$.)