3.20. To construct the geometric locus of points equidistant from the point $A(4 ; 1)$ and from the y-axis.
Solution. From the definition of a parabola, we conclude that the sought geometric locus of points is a parabola. Let's choose a coordinate system in such a way that the equation of the parabola has a canonical form. The origin should be at the midpoint of the perpendicular dropped from the focus to the directrix, i.e., at the point $O^{\prime}(2 ; 1)$. The coordinates $X^{\prime} O^{\prime} Y^{\prime}$ are related to the coordinates $X O Y$ as follows: $x^{\prime}=x-2, y^{\prime}=y-1$. The parameter $p$ is equal to the distance from the focus to the directrix: $p=4$. The equation of the parabola in the coordinate system $X^{\prime} O^{\prime} Y^{\prime}\left(y^{\prime}\right)^{2}=8 x^{\prime}$. Transitioning to the old coordinates $X O Y$, we get $(y-1)^{2}=8(x-2)$.